Purication of quantum stateFrom Wikipedia, the free encyclopedia

Contents

1 3D projection 11.1 Orthographic projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Weak perspective projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Perspective projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Pairing 62.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Pairings in cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Slightly dierent usages of the notion of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Partial trace 83.1 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.1 Invariant denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 Category theoretic notion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Partial trace for operators on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.1 Computing the partial trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Partial trace and invariant integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Partial trace as a quantum operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4.1 Comparison with classical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Peetres inequality 124.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Permanent 135.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Properties and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

i

ii CONTENTS

5.2.1 Cycle covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2.2 Perfect matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.3 Permanents of (0,1) matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.4 Van der Waerdens conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.5 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.6 MacMahons Master Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.7 Permanents of rectangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.7.1 Systems of distinct representatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Polarization identity 206.1 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6.1.1 For vector spaces with real scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.1.2 For vector spaces with complex scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.1.3 Multiple special cases for the Euclidean norm . . . . . . . . . . . . . . . . . . . . . . . . 21

6.2 Application to dot products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2.1 Relation to the law of cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.3.1 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.3.2 Symmetric bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.3.3 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.3.4 Homogeneous polynomials of higher degree . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.4 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7 Polynomial basis 257.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.3 Squaring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.4 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.5 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8 Productive matrix 288.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.2 Explicit denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

CONTENTS iii

8.4 Properties[2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.4.1 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.4.2 Transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

8.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9 Projection (linear algebra) 319.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

9.1.1 Orthogonal projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.1.2 Oblique projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9.2 Properties and classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.2.1 Orthogonal projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.2.2 Oblique projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

9.3 Canonical forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.4 Projections on normed vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.5 Applications and further considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

10 Projection-valued measure 3910.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910.2 Extensions of projection-valued measures, integrals and the spectral theorem . . . . . . . . . . . . 4010.3 Structure of projection-valued measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.4 Application in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

11 Projectivization 4311.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.2 Projective completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4311.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

12 Pseudoscalar 4412.1 Pseudoscalars in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

12.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.2 Pseudoscalars in geometric algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

13 Pseudovector 4613.1 Physical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.2 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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13.2.1 Behavior under addition, subtraction, scalar multiplication . . . . . . . . . . . . . . . . . . 4813.2.2 Behavior under cross products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

13.3 The right-hand rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.4 Formalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.5 Geometric algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

13.5.1 Transformations in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.5.2 Note on usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

13.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5113.7 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5213.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

14 Purication of quantum state 5314.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

14.1.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.2 An application: Stinesprings theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 55

14.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5514.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5614.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Chapter 1

3D projection

3D projection is any method of mapping three-dimensional points to a two-dimensional plane. As most currentmethods for displaying graphical data are based on planar( pixel information from several bitplanes) two-dimensionalmedia, the use of this type of projection is widespread, especially in computer graphics, engineering and drafting.

1.1 Orthographic projectionMain article: Orthographic projection

When the human eye looks at a scene, objects in the distance appear smaller than objects close by. Orthographicprojection ignores this eect to allow the creation of to-scale drawings for construction and engineering.Orthographic projections are a small set of transforms often used to show prole, detail or precise measurements of athree dimensional object. Common names for orthographic projections include plane, cross-section, birds-eye, andelevation.If the normal of the viewing plane (the camera direction) is parallel to one of the primary axes (which is the x, y, orz axis), the mathematical transformation is as follows; To project the 3D point ax , ay , az onto the 2D point bx , byusing an orthographic projection parallel to the y axis (prole view), the following equations can be used:

bx = sxax + cx

by = szaz + cz

where the vector s is an arbitrary scale factor, and c is an arbitrary oset. These constants are optional, and can beused to properly align the viewport. Using matrix multiplication, the equations become:

bxby

=

sx 0 00 0 sz

24axayaz

35+ cxcz

While orthographically projected images represent the three dimensional nature of the object projected, they do notrepresent the object as it would be recorded photographically or perceived by a viewer observing it directly. Inparticular, parallel lengths at all points in an orthographically projected image are of the same scale regardless ofwhether they are far away or near to the virtual viewer. As a result, lengths near to the viewer are not foreshortenedas they would be in a perspective projection.

1.2 Weak perspective projectionA weak perspective projection uses the same principles of an orthographic projection, but requires the scaling factorto be specied, thus ensuring that closer objects appear bigger in the projection, and vice versa. It can be seen as a

1

2 CHAPTER 1. 3D PROJECTION

hybrid between an orthographic and a perspective projection, and described either as a perspective projection withindividual point depths Zi replaced by an average constant depth Zave ,[1] or simply as an orthographic projectionplus a scaling.[2]

The weak-perspective model thus approximates perspective projection while using a simpler model, similar to thepure (unscaled) orthographic perspective. It is a reasonable approximation when the depth of the object along theline of sight is small compared to the distance from the camera, and the eld of view is small. With these conditions,it can be assumed that all points on a 3D object are at the same distance Zave from the camera without signicanterrors in the projection (compared to the full perspective model).

1.3 Perspective projectionSee also: Transformation matrix and Camera matrix

When the human eye views a scene, objects in the distance appear smaller than objects close by - this is known asperspective. While orthographic projection ignores this eect to allow accurate measurements, perspective projectionshows distant objects as smaller to provide additional realism.The perspective projection requires a more involved denition as compared to orthographic projections. A conceptualaid to understanding the mechanics of this projection is to imagine the 2D projection as though the object(s) are beingviewed through a camera viewnder. The cameras position, orientation, and eld of view control the behavior of theprojection transformation. The following variables are dened to describe this transformation:

ax;y;z - the 3D position of a point A that is to be projected. cx;y;z - the 3D position of a point C representing the camera. x;y;z - The orientation of the camera (represented by TaitBryan angles). ex;y;z - the viewers position relative to the display surface [3] which goes through point C representing thecamera.

Which results in:

bx;y - the 2D projection of a .

When cx;y;z = h0; 0; 0i; and x;y;z = h0; 0; 0i; the 3D vector h1; 2; 0i is projected to the 2D vector h1; 2i .Otherwise, to compute bx;y we rst dene a vector dx;y;z as the position of point A with respect to a coordinatesystem dened by the camera, with origin in C and rotated by with respect to the initial coordinate system. This isachieved by subtracting c from a and then applying a rotation by to the result. This transformation is often calleda camera transform, and can be expressed as follows, expressing the rotation in terms of rotations about the x, y,and z axes (these calculations assume that the axes are ordered as a left-handed system of axes): [4] [5]

24dxdydz

35 =241 0 00 cos(x) sin(x)0 sin(x) cos(x)

3524 cos(y) 0 sin(y)0 1 0 sin(y) 0 cos(y)

3524cos(z) sin(z) 0sin(z) cos(z) 00 0 1

350@24axayaz

3524cxcycz

351AThis representation corresponds to rotating by three Euler angles (more properly, TaitBryan angles), using the xyzconvention, which can be interpreted either as rotate about the extrinsic axes (axes of the scene) in the order z, y,x (reading right-to-left)" or rotate about the intrinsic axes (axes of the camera) in the order x, y, z (reading left-to-right)". Note that if the camera is not rotated ( x;y;z = h0; 0; 0i ), then the matrices drop out (as identities), and thisreduces to simply a shift: d = a c:Alternatively, without using matrices (lets replace (a-c) with x and so on, and abbreviate cos to c and sin to s):

dx = cy(szy+ czx) syzdy = sx(cyz+ sy(szy+ czx)) + cx(czy szx)dz = cx(cyz+ sy(szy+ czx)) sx(czy szx)

1.3. PERSPECTIVE PROJECTION 3

This transformed point can then be projected onto the 2D plane using the formula (here, x/y is used as the projectionplane; literature also may use x/z):[6]

bx = ezdz dx exby = ezdz dy ey

:

Or, in matrix form using homogeneous coordinates, the system

2664fxfyfzfw

3775 =26641 0 exez 00 1 eyez 00 0 1 00 0 1/ez 0

37752664dxdydz1

3775

in conjunction with an argument using similar triangles, leads to division by the homogeneous coordinate, giving

bx = fx/fwby = fy/fw :

The distance of the viewer from the display surface, ez , directly relates to the eld of view, where = 2tan1(1/ez)is the viewed angle. (Note: This assumes that you map the points (1,1) and (1,1) to the corners of your viewingsurface)The above equations can also be rewritten as:

bx = (dxsx)/(dzrx)rzby = (dysy)/(dzry)rz :

In which sx;y is the display size, rx;y is the recording surface size (CCD or lm), rz is the distance from the recordingsurface to the entrance pupil (camera center), and dz is the distance, from the 3D point being projected, to the entrancepupil.Subsequent clipping and scaling operations may be necessary to map the 2D plane onto any particular display media.

4 CHAPTER 1. 3D PROJECTION

1.4 Diagram

Bx

Ax

Bz Az

To determine which screen x-coordinate corresponds to a point at Ax; Az multiply the point coordinates by:

Bx = AxBzAz

where

Bx is the screen x coordinateAx is the model x coordinateBz is the focal lengththe axial distance from the camera center to the image planeAz is the subject distance.

Because the camera is in 3D, the same works for the screen y-coordinate, substituting y for x in the above diagramand equation.

1.5 See also 3D computer graphics Camera matrix Computer graphics Graphics card Homography Homogeneous coordinates Perspective (graphical)

1.6. REFERENCES 5

Texture mapping Virtual globe Transform and lighting

1.6 References[1] Subhashis Banerjee (2002-02-18). The Weak-Perspective Camera.

[2] Alter, T. D. (July 1992). 3D Pose from 3 Corresponding Points underWeak-Perspective Projection (PDF) (Technical report).MIT AI Lab.

[3] Ingrid Carlbom, Joseph Paciorek (1978). Planar Geometric Projections and Viewing Transformations (PDF). ACMComputing Surveys 10 (4): 465502. doi:10.1145/356744.356750..

[4] Riley, K F (2006). Mathematical Methods for Physics and Engineering. Cambridge University Press. pp. 931, 942.doi:10.2277/0521679710. ISBN 0-521-67971-0.

[5] Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, Mass.: Addison-Wesley Pub. Co. pp. 146148. ISBN0-201-02918-9.

[6] Sonka, M; Hlavac, V; Boyle, R (1995). Image Processing, Analysis & Machine Vision (2nd ed.). Chapman and Hall. p.14. ISBN 0-412-45570-6.

1.7 External links A case study in camera projection Creating 3D Environments from Digital Photographs

1.8 Further reading Kenneth C. Finney (2004). 3D Game Programming All in One. Thomson Course. p. 93. ISBN 978-1-59200-136-1.

Koehler; Dr. Ralph. 2D/3D Graphics and Splines with Source Code. ISBN 0759611874.

Chapter 2

Pairing

This article is about the mathematics concept. For other uses, see Pair (disambiguation).

The concept of pairing treated here occurs in mathematics.

2.1 Denition

Let R be a commutative ring with unity, and let M, N and L be three R-modules.A pairing is any R-bilinear map e : M N ! L . That is, it satises

e(rm; n) = e(m; rn) = re(m;n)

e(m1 +m2; n) = e(m1; n) + e(m2; n) and e(m;n1 + n2) = e(m;n1) + e(m;n2)

for any r 2 R and anym;m1;m2 2M and any n; n1; n2 2 N . Or equivalently, a pairing is an R-linear map

M R N ! L

whereM R N denotes the tensor product of M and N.A pairing can also be considered as an R-linear map : M ! HomR(N;L) , which matches the rst denition bysetting (m)(n) := e(m;n) .A pairing is called perfect if the above map is an isomorphism of R-modules.If N = M a pairing is called alternating if for the above map we have e(m;m) = 0 .A pairing is called non-degenerate if for the above map we have that e(m;n) = 0 for allm implies n = 0 .

2.2 Examples

Any scalar product on a real vector space V is a pairing (set M = N = V, R = R in the above denitions).The determinant map (2 2 matrices over k) k can be seen as a pairing k2 k2 ! k .The Hopf map S3 ! S2 written as h : S2 S2 ! S2 is an example of a pairing. In [1] for instance, Hardie et al.present an explicit construction of the map using poset models.

6

2.3. PAIRINGS IN CRYPTOGRAPHY 7

2.3 Pairings in cryptographyMain article: Pairing-based cryptography

In cryptography, often the following specialized denition is used:[2]

Let G1; G2 be additive groups and GT a multiplicative group, all of prime order p . Let P 2 G1; Q 2 G2 begenerators of G1 and G2 respectively.A pairing is a map: e : G1 G2 ! GTfor which the following holds:

1. Bilinearity: 8a; b 2 Zp : eP a; Qb

= e (P;Q)

ab

2. Non-degeneracy: e (P;Q) 6= 13. For practical purposes, e has to be computable in an ecient manner

Note that is also common in cryptographic literature for all groups to be written in multiplicative notation.In cases when G1 = G2 = G , the pairing is called symmetric. If, furthermore, G is cyclic, the map e will becommutative; that is, for any P;Q 2 G , we have e(P;Q) = e(Q;P ) . This is because for a generator g 2 G ,there exist integers p , q such that P = gp and Q = gq . Therefore e(P;Q) = e(gp; gq) = e(g; g)pq = e(gq; gp) =e(Q;P ) .The Weil pairing is an important pairing in elliptic curve cryptography; e.g., it may be used to attack certain ellipticcurves (see MOV attack). It and other pairings have been used to develop identity-based encryption schemes.

2.4 Slightly dierent usages of the notion of pairingScalar products on complex vector spaces are sometimes called pairings, although they are not bilinear. For example,in representation theory, one has a scalar product on the characters of complex representations of a nite group whichis frequently called character pairing.

2.5 References[1] A nontrivial pairing of nite T0 spaces Authors: Hardie K.A.1; Vermeulen J.J.C.; Witbooi P.J. Source: Topology and its

Applications, Volume 125, Number 3, 20 November 2002 , pp. 533-542(10)

[2] Dan Boneh, Matthew K. Franklin, Identity-Based Encryption from the Weil Pairing Advances in Cryptology - Proceedingsof CRYPTO 2001 (2001)

2.6 External links The Pairing-Based Crypto Lounge

Chapter 3

Partial trace

Left hand side shows a full density matrix AB of a bipartite qubit system. The partial trace is performed over a subsystem of 2 by2 dimension (single qubit density matrix) . The right hand side shows the resulting 2 by 2 reduced density matrix A .

In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is ascalar valued function on operators, the partial trace is an operator-valued function. The partial trace has applicationsin quantum information and decoherence which is relevant for quantum measurement and thereby to the decoherentapproaches to interpretations of quantummechanics, including consistent histories and the relative state interpretation.

3.1 DetailsSuppose V , W are nite-dimensional vector spaces over a eld, with dimensions m and n , respectively. For anyspace A let L(A) denote the space of linear operators on A . The partial trace overW , TrW : L(V W ) ! L(V ), is a mapping

T 2 L(V W ) 7! TrW (T ) 2 L(V )

It is dened as follows: let

e1; : : : ; em

and

f1; : : : ; fn

be bases for V andW respectively; then T has a matrix representation

8

3.1. DETAILS 9

fak`;ijg 1 k; i m; 1 `; j nrelative to the basis

ek f`of

V WNow for indices k, i in the range 1, ..., m, consider the sum

bk;i =nXj=1

akj;ij :

This gives a matrix bk, i. The associated linear operator onV is independent of the choice of bases and is by denitionthe partial trace.Among physicists, this is often called tracing out or tracing overW to leave only an operator on V in the contextwhereW and V are Hilbert spaces associated with quantum systems (see below).

3.1.1 Invariant denitionThe partial trace operator can be dened invariantly (that is, without reference to a basis) as follows: it is the uniquelinear operator

TrW : L(V W ) ! L(V )such that

TrW (R S) = R Tr(S) 8R 2 L(V ) 8S 2 L(W ):To see that the conditions above determine the partial trace uniquely, let v1; : : : ; vm form a basis forV , letw1; : : : ; wnform a basis forW , let Eij : V ! V be the map that sends vi to vj (and all other basis elements to zero), and letFkl : W ! W be the map that sends wk to wl . Since the vectors vi wk form a basis for V W , the mapsEij Fkl form a basis for L(V W ) .From this abstract denition, the following properties follow:

TrW (IVW ) = dimW IVTrW (T (IV S)) = TrW ((IV S)T ) 8S 2 L(W ) 8T 2 L(V W ):

3.1.2 Category theoretic notionIt is the partial trace of linear transformations that is the subject of Joyal, Street, and Veritys notion of Tracedmonoidal category. A traced monoidal category is a monoidal category (C;; I) together with, for objects X, Y, Uin the category, a function of Hom-sets,

TrUX;Y : HomC(X U; Y U) ! HomC(X;Y )satisfying certain axioms.Another case of this abstract notion of partial trace takes place in the category of nite sets and bijections betweenthem, in which the monoidal product is disjoint union. One can show that for any nite sets, X,Y,U and bijectionX + U = Y + U there exists a corresponding partially traced bijectionX = Y .

10 CHAPTER 3. PARTIAL TRACE

3.2 Partial trace for operators on Hilbert spacesThe partial trace generalizes to operators on innite dimensional Hilbert spaces. Suppose V, W are Hilbert spaces,and let

ffigi2Ibe an orthonormal basis forW. Now there is an isometric isomorphism

M`2I

(V Cf`) ! V W

Under this decomposition, any operator T 2 L(V W ) can be regarded as an innite matrix of operators on V

26666664T11 T12 : : : T1j : : :T21 T22 : : : T2j : : :... ... ...

Tk1 Tk2 : : : Tkj : : :... ... ...

37777775;

where Tk` 2 L(V ) .First suppose T is a non-negative operator. In this case, all the diagonal entries of the above matrix are non-negativeoperators on V. If the sum

X`

T``

converges in the strong operator topology of L(V), it is independent of the chosen basis of W. The partial traceTrW(T) is dened to be this operator. The partial trace of a self-adjoint operator is dened if and only if the partialtraces of the positive and negative parts are dened.

3.2.1 Computing the partial traceSupposeW has an orthonormal basis, which we denote by ket vector notation as fj`ig` . Then

TrW

0@Xk;`

Tk` jkih`j1A =X

j

Tjj :

3.3 Partial trace and invariant integrationIn the case of nite dimensional Hilbert spaces, there is a useful way of looking at partial trace involving integrationwith respect to a suitably normalized Haar measure over the unitary group U(W) ofW. Suitably normalized meansthat is taken to be a measure with total mass dim(W).Theorem. Suppose V,W are nite dimensional Hilbert spaces. Then

ZU(W )

(IV U)T (IV U) d(U)

commutes with all operators of the form IV S and hence is uniquely of the form R IW . The operator R is thepartial trace of T.

3.4. PARTIAL TRACE AS A QUANTUM OPERATION 11

3.4 Partial trace as a quantum operationThe partial trace can be viewed as a quantum operation. Consider a quantum mechanical system whose state space isthe tensor productHAHB of Hilbert spaces. Amixed state is described by a density matrix , that is a non-negativetrace-class operator of trace 1 on the tensor product HA HB : The partial trace of with respect to the system B,denoted by A , is called the reduced state of on system A. In symbols,

A = TrB :

To show that this is indeed a sensible way to assign a state on the A subsystem to , we oer the following justication.Let M be an observable on the subsystem A, then the corresponding observable on the composite system isM I. However one chooses to dene a reduced state A , there should be consistency of measurement statistics. Theexpectation value of M after the subsystem A is prepared in A and that of M I when the composite system isprepared in should be the same, i.e. the following equality should hold:

Tr(M A) = Tr(M I ):

We see that this is satised if A is as dened above via the partial trace. Furthermore it is the unique such operation.Let T(H) be the Banach space of trace-class operators on the Hilbert space H. It can be easily checked that the partialtrace, viewed as a map

TrB : T (HA HB) ! T (HA)

is completely positive and trace-preserving.The partial trace map as given above induces a dual map TrB between the C*-algebras of bounded operators on HAandHA HB given by

TrB(A) = A I:

TrB maps observables to observables and is the Heisenberg picture representation of TrB .

3.4.1 Comparison with classical caseSuppose instead of quantum mechanical systems, the two systems A and B are classical. The space of observablesfor each system are then abelian C*-algebras. These are of the form C(X) and C(Y) respectively for compact spacesX, Y. The state space of the composite system is simply

C(X) C(Y ) = C(X Y ):

A state on the composite system is a positive element of the dual of C(X Y), which by the Riesz-Markov theoremcorresponds to a regular Borel measure on X Y. The corresponding reduced state is obtained by projecting themeasure to X. Thus the partial trace is the quantum mechanical equivalent of this operation.

Chapter 4

Peetres inequality

In mathematics, Peetres inequality, named after Jaak Peetre, says that for any real number t and any vectors x andy in Rn, the following inequality holds:

1 + jxj21 + jyj2

t 2jtj(1 + jx yj2)jtj:

4.1 References Chazarain, J.; Piriou, A. (2011), Introduction to the Theory of Linear Partial Dierential Equations, Studies inMathematics and its Applications, Elsevier, p. 90, ISBN 9780080875354.

Ruzhansky, Michael; Turunen, Ville (2009), Pseudo-Dierential Operators and Symmetries: Background Anal-ysis and Advanced Topics, Pseudo-Dierential Operators, Theory and Applications 2, Springer, p. 321, ISBN9783764385132.

Saint Raymond, Xavier (1991), Elementary Introduction to the Theory of Pseudodierential Operators, Studiesin Advanced Mathematics 3, CRC Press, p. 21, ISBN 9780849371585.

This article incorporates material from Peetres inequality on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

12

Chapter 5

Permanent

For other uses, see Permanent (disambiguation).

In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. Thepermanent, as well as the determinant, is a polynomial in the entries of the matrix.[1] Both permanent and determinantare special cases of a more general function of a matrix called the immanant.

5.1 Denition

The permanent of an n-by-n matrix A = (ai,j) is dened as

perm(A) =X2Sn

nYi=1

ai;(i):

The sum here extends over all elements of the symmetric group Sn; i.e. over all permutations of the numbers 1, 2,..., n.For example,

perma bc d

= ad+ bc;

and

perm

0@a b cd e fg h i

1A = aei+ bfg + cdh+ ceg + bdi+ afh:The denition of the permanent ofA diers from that of the determinant ofA in that the signatures of the permutationsare not taken into account.The permanent of a matrix A is denoted per A, perm A, or Per A, sometimes with parentheses around the argument.In his monograph, Minc (1984) uses Per(A) for the permanent of rectangular matrices, and uses per(A) when A is asquare matrix. Muir (1882) uses the notation

+

j+

j .The word, permanent, originated with Cauchy in 1812 as fonctions symtriques permanentes for a related type offunction,[2] and was used by Muir (1882) in the modern, more specic, sense.[3]

13

14 CHAPTER 5. PERMANENT

5.2 Properties and applicationsIf one views the permanent as a map that takes n vectors as arguments, then it is a multilinear map and it is symmetric(meaning that any order of the vectors results in the same permanent). Furthermore, given a square matrixA = (aij)of order n, we have:[4]

perm(A) is invariant under arbitrary permutations of the rows and/or columns of A. This property may bewritten symbolically as perm(A) = perm(PAQ) for any appropriately sized permutation matrices P and Q,

multiplying any single row or column of A by a scalar s changes perm(A) to sperm(A), perm(A) is invariant under transposition, that is, perm(A) = perm(A).

If A = (aij) and B = (bij) are square matrices of order n then,[5]

perm(A+B) =Xs;t

perm(aij)i2s;j2t perm(bij)i2s;j2t;

where s and t are subsets of the same size of {1,2,...,n} and s; t are their respective complements in that set.On the other hand, the basic multiplicative property of determinants is not valid for permanents.[6] A simple exampleshows that this is so.

4 = perm1 11 1

perm

1 11 1

6= perm

1 11 1

1 11 1

= perm

2 22 2

= 8:

A formula similar to Laplaces for the development of a determinant along a row, column or diagonal is also valid forthe permanent;[7] all signs have to be ignored for the permanent. For example, expanding along the rst column,

perm

0BB@1 1 1 12 1 0 03 0 1 04 0 0 1

1CCA = 1perm0@1 0 00 1 00 0 1

1A+2perm0@1 1 10 1 00 0 1

1A+3perm0@1 1 11 0 00 0 1

1A+4perm0@1 1 11 0 00 1 0

1A = 1(1)+2(1)+3(1)+4(1) = 10;while expanding along the last row gives,

perm

0BB@1 1 1 12 1 0 03 0 1 04 0 0 1

1CCA = 4perm0@1 1 11 0 00 1 0

1A+0perm0@1 1 12 0 03 1 0

1A+0perm0@1 1 12 1 03 0 0

1A+1perm0@1 1 12 1 03 0 1

1A = 4(1)+0+0+1(6) = 10:Unlike the determinant, the permanent has no easy geometrical interpretation; it is mainly used in combinatorics andin treating boson Greens functions in quantum eld theory. However, it has two graph-theoretic interpretations: asthe sum of weights of cycle covers of a directed graph, and as the sum of weights of perfect matchings in a bipartitegraph.

5.2.1 Cycle coversAny square matrix A = (aij) can be viewed as the adjacency matrix of a weighted directed graph, with aij repre-senting the weight of the arc from vertex i to vertex j. A cycle cover of a weighted directed graph is a collection ofvertex-disjoint directed cycles in the digraph that covers all vertices in the graph. Thus, each vertex i in the digraphhas a unique successor (i) in the cycle cover, and is a permutation on f1; 2; : : : ; ng where n is the number ofvertices in the digraph. Conversely, any permutation on f1; 2; : : : ; ng corresponds to a cycle cover in which thereis an arc from vertex i to vertex (i) for each i.If the weight of a cycle-cover is dened to be the product of the weights of the arcs in each cycle, then

5.3. PERMANENTS OF (0,1) MATRICES 15

Weight() =nYi=1

ai;(i):

The permanent of an n n matrix A is dened as

perm(A) =X

nYi=1

ai;(i)

where is a permutation over f1; 2; : : : ; ng . Thus the permanent of A is equal to the sum of the weights of allcycle-covers of the digraph.

5.2.2 Perfect matchingsA square matrix A = (aij) can also be viewed as the adjacency matrix of a bipartite graph which has verticesx1; x2; : : : ; xn on one side and y1; y2; : : : ; yn on the other side, with aij representing the weight of the edge fromvertex xi to vertex yj . If the weight of a perfect matching that matches xi to y(i) is dened to be the product ofthe weights of the edges in the matching, then

Weight() =nYi=1

ai;(i):

Thus the permanent of A is equal to the sum of the weights of all perfect matchings of the graph.

5.3 Permanents of (0,1) matricesThe permanents of matrices that only have 0 and 1 as entries are often the answers to certain counting questionsinvolving the structures that the matrices represent. This is particularly true of adjacency matrices in graph theoryand incidence matrices of symmetric block designs.In an unweighted, directed, simple graph (a digraph), if we set each aij to be 1 if there is an edge from vertex i tovertex j, then each nonzero cycle cover has weight 1, and the adjacency matrix has 0-1 entries. Thus the permanentof a (0,1)-matrix is equal to the number of vertex cycle covers of an unweighted directed graph.For an unweighted bipartite graph, if we set ai,j = 1 if there is an edge between the vertices xi and yj and ai,j =0 otherwise, then each perfect matching has weight 1. Thus the number of perfect matchings in G is equal to thepermanent of matrix A.[8]

Let (n,k) be the class of all (0,1)-matrices of order n with each row and column sum equal to k. Every matrix A inthis class has perm(A) > 0.[9] The incidence matrices of projective planes are in the class (n2 + n + 1, n + 1) for nan integer > 1. The permanents corresponding to the smallest projective planes have been calculated. For n = 2, 3,and 4 the values are 24, 3852 and 18,534,400 respectively.[9] Let Z be the incidence matrix of the projective planewith n = 2, the Fano plane. Remarkably, perm(Z) = 24 = |det (Z)|, the absolute value of the determinant of Z. Thisis a consequence of Z being a circulant matrix and the theorem:[10]

If A is a circulant matrix in the class (n,k) then if k > 3, perm(A) > |det (A)| and if k = 3, perm(A) =|det (A)|. Furthermore, when k = 3, by permuting rows and columns, A can be put into the form of adirect sum of e copies of the matrix Z and consequently, n = 7e and perm(A) = 24e.

Permanents can also be used to calculate the number of permutations with restricted (prohibited) positions. For thestandard n-set, {1,2,...,n}, let A = (aij) be the (0,1)-matrix where aij = 1 if i j is allowed in a permutation andaij = 0 otherwise. Then perm(A) counts the number of permutations of the n-set which satisfy all the restrictions.[7]Two well known special cases of this are the solution of the derangement problem (the number of permutations withno xed points) given by:

16 CHAPTER 5. PERMANENT

perm(J I) = perm

0BBBBB@0 1 1 : : : 11 0 1 : : : 11 1 0 : : : 1... ... ... . . . ...1 1 1 : : : 0

1CCCCCA = n!nXi=0

(1)ii!

;

where J is the all 1s matrix and I is the identity matrix, each of order n, and the solution to the mnage problemgiven by:

perm(J I I 0) = perm

0BBBBB@0 0 1 : : : 11 0 0 : : : 11 1 0 : : : 1... ... ... . . . ...0 1 1 : : : 0

1CCCCCA = 2 n!nX

k=0

(1)k 2n2n k

2n kk

(n k)!;

where I' is the (0,1)-matrix whose only non-zero entries are on the rst superdiagonal.The following result was conjectured by H. Minc in 1967[11] and proved by L. M. Brgman in 1973.[12]

Theorem: Let A be an n n (0,1)-matrix with ri ones in row i, 1 i n. Then

permA nYi=1

(ri)!1/ri :

5.4 Van der Waerdens conjectureIn 1926 Van der Waerden conjectured that the minimum permanent among all n n doubly stochastic matrices isn!/nn, achieved by the matrix for which all entries are equal to 1/n.[13] Proofs of this conjecture were published in1980 by B. Gyires[14] and in 1981 by G. P. Egorychev[15] and D. I. Falikman;[16] Egorychevs proof is an application ofthe AlexandrovFenchel inequality.[17] For this work, Egorychev and Falikman won the Fulkerson Prize in 1982.[18]

5.5 ComputationMain articles: Computing the permanent and Permanent is sharp-P-complete

The nave approach, using the denition, of computing permanents is computationally infeasible even for relativelysmall matrices. One of the fastest known algorithms is due to H. J. Ryser (Ryser (1963, p. 27)). Rysers method isbased on an inclusionexclusion formula that can be given[19] as follows: Let Ak be obtained from A by deleting kcolumns, let P (Ak) be the product of the row-sums of Ak , and let k be the sum of the values of P (Ak) over allpossible Ak . Then

perm(A) =n1Xk=0

(1)kk:

It may be rewritten in terms of the matrix entries as follows:

perm(A) = (1)nX

Sf1;:::;ng(1)jSj

nYi=1

Xj2S

aij :

5.6. MACMAHONS MASTER THEOREM 17

The permanent is believed to be more dicult to compute than the determinant. While the determinant can be com-puted in polynomial time by Gaussian elimination, Gaussian elimination cannot be used to compute the permanent.Moreover, computing the permanent of a (0,1)-matrix is #P-complete. Thus, if the permanent can be computed inpolynomial time by any method, then FP = #P, which is an even stronger statement than P = NP. When the entriesof A are nonnegative, however, the permanent can be computed approximately in probabilistic polynomial time, upto an error of M, where M is the value of the permanent and > 0 is arbitrary.[20]

5.6 MacMahons Master TheoremMain article: MacMahon Master theorem

Another way to view permanents is via multivariate generating functions. Let A = (aij) be a square matrix of ordern. Consider the multivariate generating function:

F (x1; x2; : : : ; xn) =nYi=1

0@ nXj=1

aijxj

1A =0@ nXj=1

a1jxj

1A0@ nXj=1

a2jxj

1A 0@ nXj=1

anjxj

1A :The coecient of x1x2 : : : xn in F (x1; x2; : : : ; xn) is perm(A).[21]

As a generalization, for any sequence of n non-negative integers, s1; s2; : : : ; sn dene:

perm(s1;s2;:::;sn)(A) := of coecient xs11 xs22 xsnn in0@ nXj=1

a1jxj

1As1 0@ nXj=1

a2jxj

1As2 0@ nXj=1

anjxj

1Asn :MacMahons Master Theorem relating permanents and determinants is:[22]

perm(s1;s2;:::;sn)(A) = of coecient xs11 xs22 xsnn in1

det(I XA) ;

where I is the order n identity matrix and X is the diagonal matrix with diagonal [x1; x2; : : : ; xn]:

5.7 Permanents of rectangular matricesThe permanent function can be generalized to apply to non-square matrices. Indeed, several authors make this thedenition of a permanent and consider the restriction to square matrices a special case.[23] Specically, for an m nmatrix A = (aij) with m n, dene

perm(A) =X

2P(n;m)a1(1)a2(2) : : : am(m)

where P(n,m) is the set of all m-permutations of the n-set {1,2,...,n}.[24]

Rysers computational result for permanents also generalizes. If A is an m n matrix with m n, let Ak be obtainedfrom A by deleting k columns, let P (Ak) be the product of the row-sums of Ak , and let k be the sum of the valuesof P (Ak) over all possible Ak . Then

perm(A) =Pm1k=0 (1)knm+kk nm+k: [6]

18 CHAPTER 5. PERMANENT

5.7.1 Systems of distinct representativesThe generalization of the denition of a permanent to non-square matrices allows the concept to be used in a morenatural way in some applications. For instance:Let S1, S2, ..., Sm be subsets (not necessarily distinct) of an n-set with m n. The incidence matrix of this collectionof subsets is an m n (0,1)-matrix A. The number of systems of distinct representatives (SDRs) of this collection isperm(A).[25]

5.8 See also Determinant BapatBeg theorem, an application of permanent in order statistics

5.9 Notes[1] Marcus, Marvin; Minc, Henryk (1965). Permanents. Amer. Math. Monthly 72: 577591. doi:10.2307/2313846.

[2] Cauchy, A. L. (1815), Mmoire sur les fonctions qui ne peuvent obtenir que deux valeurs gales et de signes contrairespar suite des transpositions opres entre les variables quelles renferment., Journal de l'cole Polytechnique 10: 91169

[3] van Lint & Wilson 2001, p. 108

[4] Ryser 1963, pp. 25 26

[5] Percus 1971, p. 2

[6] Ryser 1963, p. 26

[7] Percus 1971, p. 12

[8] Dexter Kozen. The Design and Analysis of Algorithms. Springer-Verlag, New York, 1991. ISBN 978-0-387-97687-7; pp.141142

[9] Ryser 1963, p. 124

[10] Ryser 1963, p. 125

[11] Minc, H. (1967), An inequality for permanents of (0,1)matrices, Journal of Combinatorial Theory 2: 321326, doi:10.1016/s0021-9800(67)80033-4

[12] van Lint & Wilson 2001, p. 101

[13] van der Waerden, B. L. (1926), Aufgabe 45, Jber. Deutsch. Math.-Verein. 35: 117.

[14] Gyires, B. (1980), The common source of several inequalities concerning doubly stochastic matrices, Publicationes Math-ematicae Institutum Mathematicum Universitatis Debreceniensis 27 (3-4): 291304, MR 604006.

[15] Egoryev, G. P. (1980), Reshenie problemy van-der-Vardena dlya permanentov (in Russian), Krasnoyarsk: Akad. NaukSSSR Sibirsk. Otdel. Inst. Fiz., p. 12, MR 602332. Egorychev, G. P. (1981), Proof of the van der Waerden conjecturefor permanents, Akademiya Nauk SSSR (in Russian) 22 (6): 6571, 225, MR 638007. Egorychev, G. P. (1981), Thesolution of van der Waerdens problem for permanents, Advances in Mathematics 42 (3): 299305, doi:10.1016/0001-8708(81)90044-X, MR 642395.

[16] Falikman, D. I. (1981), Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix,Akademiya Nauk Soyuza SSR (in Russian) 29 (6): 931938, 957, MR 625097.

[17] Brualdi (2006) p.487

[18] Fulkerson Prize, Mathematical Optimization Society, retrieved 2012-08-19.

[19] van Lint & Wilson (2001) p. 99

[20] Jerrum, M.; Sinclair, A.; Vigoda, E. (2004), A polynomial-time approximation algorithm for the permanent of a matrixwith nonnegative entries, Journal of the ACM 51: 671697, doi:10.1145/1008731.1008738

5.10. REFERENCES 19

[21] Percus 1971, p. 14

[22] Percus 1971, p. 17

[23] In particular, Minc (1984) and Ryser (1963) do this.

[24] Ryser 1963, p. 25

[25] Ryser 1963, p. 54

5.10 References Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications108. Cambridge: Cambridge University Press. ISBN 0-521-86565-4. Zbl 1106.05001.

Minc, Henryk (1978). Permanents. Encyclopedia of Mathematics and its Applications 6. With a foreword byMarvin Marcus. Reading, MA: AddisonWesley. ISSN 0953-4806. OCLC 3980645. Zbl 0401.15005.

Muir, Thomas; William H. Metzler. (1960) [1882]. A Treatise on the Theory of Determinants. New York:Dover. OCLC 535903.

Percus, J.K. (1971), Combinatorial Methods, Applied Mathematical Sciences #4, New York: Springer-Verlag,ISBN 0-387-90027-6

Ryser, Herbert John (1963), Combinatorial Mathematics, The Carus Mathematical Monographs #14, TheMathematical Association of America

van Lint, J.H.; Wilson, R.M. (2001),ACourse in Combinatorics, CambridgeUniversity Press, ISBN0521422604

5.11 Further reading Hall, Jr., Marshall (1986), Combinatorial Theory (2nd ed.), New York: John Wiley & Sons, pp. 5672, ISBN0-471-09138-3 Contains a proof of the Van der Waerden conjecture.

Marcus,M.; Minc, H. (1965), Permanents, TheAmericanMathematicalMonthly 72: 577591, doi:10.2307/2313846

5.12 External links Permanent at PlanetMath Van der Waerdens permanent conjecture at PlanetMath.org.

Chapter 6

Polarization identity

See also: Parallelogram law, Norm (mathematics) and Inner product spaceIn mathematics, the polarization identity is any one of a family of formulas that express the inner product of twovectors in terms of the norm of a normed vector space. Let kxk denote the norm of vector x and hx; yi the innerproduct of vectors x and y. Then the underlying theorem, attributed to Frchet, von Neumann and Jordan, is statedas:[1][2]

In a normed space (V, k k ), if the parallelogram law holds, then there is an inner product on V suchthat kxk2 = hx; xi for all x 2 V .

6.1 FormulaThe various forms given below are all related by the parallelogram law:

2kuk2 + 2kvk2 = ku+ vk2 + ku vk2:

The polarization identity can be generalized to various other contexts in abstract algebra, linear algebra, and functionalanalysis.

6.1.1 For vector spaces with real scalars

If V is a real vector space, then the inner product is dened by the polarization identity

hx; yi = 14

kx+ yk2 kx yk2 8 x; y 2 V :6.1.2 For vector spaces with complex scalars

If V is a complex vector space the inner product is given by the polarization identity:

hx; yi = 14

kx+ yk2 kx yk2 + ikx+ iyk2 ikx iyk2 8 x; y 2 V ;where i is the imaginary unit. Note that this denes an inner product which is linear in its rst and semilinear in itssecond argument. To adjust for contrary denition, one needs to take the complex conjugate.

20

6.1. FORMULA 21

Vectors involved in the polarization identity.

6.1.3 Multiple special cases for the Euclidean norm

A special case is an inner product given by the dot product, the so-called standard or Euclidean inner product. In thiscase, common forms of the identity include:

22 CHAPTER 6. POLARIZATION IDENTITY

u v = 12

ku+ vk2 kuk2 kvk2 ; (1)u v = 1

2

kuk2 + kvk2 ku vk2 ; (2)u v = 1

4

ku+ vk2 ku vk2 : (3)6.2 Application to dot products

6.2.1 Relation to the law of cosinesThe second form of the polarization identity can be written as

ku vk2 = kuk2 + kvk2 2(u v):This is essentially a vector form of the law of cosines for the triangle formed by the vectors u, v, and u v. Inparticular,

u v = kuk kvk cos ;where is the angle between the vectors u and v.

6.2.2 DerivationThe basic relation between the norm and the dot product is given by the equation

kvk2 = v v:Then

ku+ vk2 = (u+ v) (u+ v)= (u u) + (u v) + (v u) + (v v)= kuk2 + kvk2 + 2(u v);

and similarly

ku vk2 = kuk2 + kvk2 2(u v):Forms (1) and (2) of the polarization identity now follow by solving these equations for u v, while form (3) followsfrom subtracting these two equations. (Adding these two equations together gives the parallelogram law.)

6.3 Generalizations

6.3.1 NormsIn linear algebra, the polarization identity applies to any norm on a vector space dened in terms of an inner productby the equation

6.3. GENERALIZATIONS 23

kvk =phv; vi:

As noted for the dot product case above, for real vectors u and v, an angle can be introduced using:[3]

hu; vi = kukkvk cos ; ( < ) ;which is acceptable by virtue of the CauchySchwarz inequality:

hu; vi kukkvk :This inequality insures that the magnitude of the above dened cosine 1. The choice of the cosine function ensuresthat when hu; vi = 0 (orthogonal vectors), the angle = /2.In this case, the identities become

hu; vi = 12ku+ vk2 kuk2 kvk2 ;

hu; vi = 12kuk2 + kvk2 ku vk2 ;

hu; vi = 14ku+ vk2 ku vk2 :

Conversely, if a norm on a vector space satises the parallelogram law, then any one of the above identities can beused to dene a compatible inner product. In functional analysis, introduction of an inner product norm like this oftenis used to make a Banach space into a Hilbert space.

6.3.2 Symmetric bilinear formsThe polarization identities are not restricted to inner products. If B is any symmetric bilinear form on a vector space,and Q is the quadratic form dened by

Q(v) = B(v; v);

then

2B(u; v) = Q(u+ v)Q(u)Q(v);2B(u; v) = Q(u) +Q(v)Q(u v);4B(u; v) = Q(u+ v)Q(u v):The so-called symmetrizationmap generalizes the latter formula, replacingQ by a homogeneous polynomial of degreek dened by Q(v)=B(v,...,v), where B is a symmetric k-linear map.The formulas above even apply in the case where the eld of scalars has characteristic two, though the left-hand sidesare all zero in this case. Consequently, in characteristic two there is no formula for a symmetric bilinear form interms of a quadratic form, and they are in fact distinct notions, a fact which has important consequences in L-theory;for brevity, in this context symmetric bilinear forms are often referred to as symmetric forms.These formulas also apply to bilinear forms on modules over a commutative ring, though again one can only solvefor B(u, v) if 2 is invertible in the ring, and otherwise these are distinct notions. For example, over the integers, onedistinguishes integral quadratic forms from integral symmetric forms, which are a narrower notion.More generally, in the presence of a ring involution or where 2 is not invertible, one distinguishes -quadratic formsand -symmetric forms; a symmetric form denes a quadratic form, and the polarization identity (without a factor of 2)from a quadratic form to a symmetric form is called the symmetrization map, and is not in general an isomorphism.This has historically been a subtle distinction: over the integers it was not until the 1950s that relation between twosout (integral quadratic form) and twos in (integral symmetric form) was understood - see discussion at integralquadratic form; and in the algebraization of surgery theory, Mishchenko originally used symmetric L-groups, ratherthan the correct quadratic L-groups (as in Wall and Ranicki) - see discussion at L-theory.

24 CHAPTER 6. POLARIZATION IDENTITY

6.3.3 Complex numbersIn linear algebra over the complex numbers, it is customary to use a sesquilinear inner product, with the property thathv; ui is the complex conjugate of hu; vi . In this case the standard polarization identities only give the real part ofthe inner product:

Rehu; vi = 12ku+ vk2 kuk2 kvk2 ;

Rehu; vi = 12kuk2 + kvk2 ku vk2 ;

Rehu; vi = 14ku+ vk2 ku vk2 :

Using Imhu; vi = Rehu;ivi , the imaginary part of the inner product can be retrieved as follows:

Imhu; vi = 12ku ivk2 kuk2 kvk2 ;

Imhu; vi = 12kuk2 + kvk2 ku+ ivk2 ;

Imhu; vi = 14ku ivk2 ku+ ivk2 :

6.3.4 Homogeneous polynomials of higher degreeFinally, in any of these contexts these identities may be extended to homogeneous polynomials (that is, algebraicforms) of arbitrary degree, where it is known as the polarization formula, and is reviewed in greater detail in thearticle on the polarization of an algebraic form.The polarization identity can be stated in the following way:

hu; vi = 413X

k=0

ikku+ ikvk2:

6.4 Notes and references[1] Philippe Blanchard, Erwin Brning (2003). Proposition 14.1.2 (Frchetvon NeumannJordan)". Mathematical methods

in physics: distributions, Hilbert space operators, and variational methods. Birkhuser. p. 192. ISBN 0817642285.

[2] Gerald Teschl (2009). Theorem 0.19 (Jordanvon Neumann)". Mathematical methods in quantum mechanics: with ap-plications to Schrdinger operators. American Mathematical Society Bookstore. p. 19. ISBN 0-8218-4660-4.

[3] Francis Begnaud Hildebrand (1992). Equation 66, the natural denition. Methods of applied mathematics (Reprint ofPrentice-Hall 1965 2nd ed.). Courier Dover Publications. p. 24. ISBN 0-486-67002-3.

Chapter 7

Polynomial basis

Inmathematics, a polynomial basis is basis of a polynomial ring, viewed as a vector space over the eld of coecients,or as a free module over the ring of coecients. The most common polynomial basis is the monomial basis consistingof all monomials. Other useful polynomial bases are the Bernstein basis and the various sequences of orthogonalpolynomials.In the case of a nite extension of a nite elds, polynomial basis may also refer to a basis of the extension of theform

f1; ; : : : ; m1g ;

where is the root of a primitive polynomial of degree m equal of the degree of the extension).[1]

The set of elements of GF(pm) can then be represented as:

f0; 1; ; 2; : : : ; pm2g

using Zechs logarithms.

7.1 AdditionAddition using the polynomial basis is as simple as addition modulo p. For example, in GF(3m):

(22 + 2+ 1) + (2+ 2) = 22 + 4+ 3 mod 3 = 22 +

In GF(2m), addition is especially easy, since addition and subtraction modulo 2 are the same thing (so like termscancel out), and furthermore this operation can be done in hardware using the basic XOR logic gate.

7.2 MultiplicationMultiplication of two elements in the polynomial basis can be accomplished in the normal way of multiplication, butthere are a number of ways to speed up multiplication, especially in hardware. Using the straightforward method tomultiply two elements in GF(pm) requires up to m2 multiplications in GF(p) and up to m2 m additions in GF(p).Some of the methods for reducing these values include:

Lookup tables a prestored table of results; mainly used for small elds, otherwise the table is too large toimplement

25

26 CHAPTER 7. POLYNOMIAL BASIS

The Karatsuba algorithm repeatedly breaking the multiplication into pieces, decreasing the total numberof multiplications but increasing the number of additions. As seen above, addition is very simple, but theoverhead in breaking down and recombining the parts make it prohibitive for hardware, although it is oftenused in software. It can even be used for general multiplication, and is done in many computer algebra systems.

Linear feedback shift register-based multiplication Subeld computations breaking the multiplication in GF(pm) to multiplications in GF(px) and GF(py), wherex y = m. This is not frequently used for cryptographic purposes, since some composite degree elds areavoided because of known attacks on them.

Pipelined multipliers storing intermediate results in buers so that new values can be loaded into the mul-tiplier faster

Systolic multipliers using many cells that communicate with neighboring cells only; typically systolic de-vices are used for computation-intensive operations where input and output sizes are not as important, such asmultiplication.

7.3 SquaringSquaring is an important operation because it can be used for general exponentiation as well as inversion of an element.The most basic way to square an element in the polynomial basis would be to apply a chosen multiplication algorithmon an element twice. In general case, there are minor optimizations that can be made, specically related to thefact that when multiplying an element by itself, all the bits will be the same. In practice, however, the irreduciblepolynomial for the eld is chosen with very few nonzero coecients which makes squaring in polynomial basis ofGF(2m) much simpler than multiplication.[2]

7.4 InversionInversion of elements can be accomplished in many ways, including:

Lookup tables once again, only for small elds otherwise the table is too large for implementation Subeld inversion by solving systems of equations in subelds Repeated square and multiply for example in GF(2m), A1 = A2m 2

The Extended Euclidean algorithm The Itoh-Tsujii inversion algorithm

7.5 UsageThe polynomial basis is frequently used in cryptographic applications that are based on the discrete logarithm problemsuch as elliptic curve cryptography.The advantage of the polynomial basis is that multiplication is relatively easy. For contrast, the normal basis is analternative to the polynomial basis and it has more complex multiplication but squaring is very simple. Hardwareimplementations of polynomial basis arithmetic usually consume more power than their normal basis counterparts.

7.6 References[1] Roman, Steven (1995). Field Theory. New York: Springer-Verlag. ISBN 0-387-94407-9.

[2] Huapeng, Wu (2001). Selected Areas in Cryptography: 7th Annual International Workshop, SAC 2000, Waterloo, On-tario, Canada, August 1415, 2000,. Springer. p. 118. |chapter= ignored (help)

7.7. SEE ALSO 27

7.7 See also normal basis dual basis change of basis

Chapter 8

Productive matrix

In linear algebra, a square nonnegative matrix A of order n is said to be productive, or to be a Leontief matrix, ifthere exists a n 1 nonnegative column matrix P such as P AP is a positive matrix.

8.1 HistoryThe concept of productive matrix was developed by the economist Wassily Leontief (Nobel Prize in Economics in1973) in order to model and analyze the relations between the dierent sectors of an economy.[1] The interdependencylinkages between the latter can be examined by the input-output model with empirical data.

8.2 Explicit denitionThe matrix A 2 Mn(R) is productive if and only if A > 0 and 9P 2 Mn;1(R); P > 0 such as P AP > 0 .

8.3 Examples

The matrix A =

0@ 0 1 00 1/2 1/21/4 1/2 0

1A is productive.8a 2 R+ , the matrix A =

0 a0 0

is productive since P =

a+ 11

veries the inequalities of denition.

8.4 Properties[2]

8.4.1 CharacterizationTheorem A nonnegative matrix A 2 Mn(R) is productive if and only if In A is invertible with a nonnegativeinverse.DemonstrationDirect involvement :

U 2 Mn;1(R); P > 0P = (In A)1UP AP = (In A)P = (In A)(In A)1U = U

28

8.4. PROPERTIES[2] 29

P AP > 0A

Reciprocal involvement :

We shall proceed by reductio ad absurdum.Let us assume 9P > 0 such as V = P AP > 0 & In A is singular.The endomorphism canonically associated with InA can not be injective by singularity of the matrix.Thus 9Z 2 Mn;1(R) non zero such as (In A)Z = 0 .The matrixZ veries the same properties as Z , therefore we can choose Z as an element of the kernelwith at least one positive entry;Hence c = supi2[j1;nj] zipi is nonnegative and reached with at least one value k 2 [j1; nj] .By denition of V and of Z , we can infer that:

cvk = c(pk nXi=1

akipi) = cpk nXi=1

akicpi

cpk = zk =nXi=1

akizi

Thus cvk =Pn

i=1 aki(zj cpj) 0 .Yet we know that c > 0 and that vk > 0 .Therefore there is a contradiction, ipso facto In A is necessarily invertible.Now let us assume In A is invertible but with at least one negative entry in its inverse.Hence 9X 2 Mn;1(R); X > 0 such as there is at least one negative entry in Y = (In A)1X .Then c = supi2[j1;nj] yipi is positive and reached with at least one value k 2 [j1; nj] .By denition of V and of X , we can infer that:

cvk = c(pk nXi=1

akipi) = yk nXi=1

akicpi

xk = yk nXi=1

akiyi

cvk + xk = nXi=1

aki(cpi + yi)

Thus xk cvk < 0 since 8i 2 [j1; nj]; aki > 0; cpi + yi > 0 .Yet we know that X > 0 .Therefore ther is a contradiction, ipso facto (In A)1 is necessarily nonnegative.

8.4.2 TranspositionProposition The transpose of a productive matrix is productive.Demonstration

A 2 Mn(R)(In A)1((In tA))1 = (t(In A))1 =t ((In A)1)(In tA)tA

30 CHAPTER 8. PRODUCTIVE MATRIX

8.5 ApplicationMain article: Input-output analysis

With a matrix approach of the input-output model, the consomption matrix is productive if it is economically viableand if the latter and the demand vector are nonnegative.

8.6 References[1] Kim Minju, Leontief Input-Output Model (Application of Linear Algebra to Economics)

[2] (fr)Philippe Michel, 9.2 Matrices productives, Cours de Mathmatiques pour Economistes, dition Economica, 1984

Chapter 9

Projection (linear algebra)

Orthogonal projection redirects here. For the technical drawing concept, see Orthographic projection. For a con-crete discussion of orthogonal projections in nite-dimensional linear spaces, see Vector projection.Various visualizations of the icosahedron

mPv

Pu w = Pw Px

v

u

x

The transformation P is the orthogonal projection onto the line m.

31

32 CHAPTER 9. PROJECTION (LINEAR ALGEBRA)

perspective

Net

Orthogonal

Petrie

Schlegel

Vertex gure

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself suchthat P2 = P. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once(idempotent). It leaves its image unchanged.[1] Though abstract, this denition of projection formalizes and gen-eralizes the idea of graphical projection. One can also consider the eect of a projection on a geometrical object byexamining the eect of the projection on points in the object.

9.1 Simple example

9.1.1 Orthogonal projectionFor example, the function which maps the point (x, y, z) in three-dimensional space R3 to the point (x, y, 0) is anorthogonal projection onto the xy plane. This function is represented by the matrix

P =

241 0 00 1 00 0 0

35:

9.2. PROPERTIES AND CLASSIFICATION 33

The action of this matrix on an arbitrary vector is

P

0@xyz

1A =0@xy0

1A:To see that P is indeed a projection, i.e., P = P2, we compute

P 2

0@xyz

1A = P0@xy0

1A =0@xy0

1A = P0@xyz

1A:9.1.2 Oblique projectionA simple example of a non-orthogonal (oblique) projection (for denition see below) is

P =

0 0 1

:

Via matrix multiplication, one sees that

P 2 =

0 0 1

0 0 1

=

0 0 1

= P:

proving that P is indeed a projection.The projection P is orthogonal if and only if = 0.

9.2 Properties and classicationLetW be a nite dimensional vector space and P be a projection onW. Suppose the subspaces U and V are the rangeand kernel of P respectively. Then P has the following basic properties:

1. By denition, P is idempotent (i.e. P2 = P).2. P is the identity operator I on U

8x 2 U : Px = x3. We have a direct sumW = U V. Every vector x inW may be decomposed uniquely as x = u + v with u = Px

and v = x Px = (I P)x, and where u is in U and v is in V.

The range and kernel of a projection are complementary, as are P and Q = I P. The operator Q is also a projectionand the range and kernel of P become the kernel and range of Q and vice versa. We say P is a projection along Vonto U (kernel/range) and Q is a projection along U onto V.In innite dimensional vector spaces spectrum of a projection is contained in {0, 1}, as

(I P )1 = 1I +

1

( 1)P

Only 0 and 1 can be an eigenvalue of a projection. The corresponding eigenspaces are (respectively) the kernel andrange of the projection. Decomposition of a vector space into direct sums is not unique in general. Therefore, givena subspace V, there may be many projections whose range (or kernel) is V.If a projection is nontrivial it has minimal polynomial X2 X = X(X I), which factors into distinct roots, and thusP is diagonalizable.The product of projections is not, in general, a projection, even if they are orthogonal. If projections commute, thentheir product is a projection.

34 CHAPTER 9. PROJECTION (LINEAR ALGEBRA)

v

u

x

k

m

Tv

Tu

w = Tw

Tx

The transformation T is the projection along k onto m. The range of T is m and the null space is k.

9.2.1 Orthogonal projectionsWhen the vector space W has an inner product and is complete (is a Hilbert space) the concept of orthogonalitycan be used. An orthogonal projection is a projection for which the range U and the null space V are orthogonalsubspaces. Thus, for every x and y inW, hPx; (y Py)i = h(x Px); Pyi = 0 . Equivalently:

hx; Pyi = hPx; Pyi = hPx; yiA projection is orthogonal if and only if it is self-adjoint. Using the self-adjoint and idempotent properties of P, forany x and y inW we have Px U, y Py V, and

hPx; y Pyi = hP 2x; y Pyi = hPx; P (I P )yi = hPx; (P P 2)yi = 0where h; i is the inner product associated with W. Therefore, Px and y Py are orthogonal.[2] The other direction,namely that if P is orthogonal then it is self-adjoint, follows from

hx; Pyi = hPx; yi = hx; P yi

9.2. PROPERTIES AND CLASSIFICATION 35

for every x and y inW; thus P = P*.

Properties and special cases

An orthogonal projection is a bounded operator. This is because for every v in the vector space we have, by CauchySchwarz inequality:

kPvk2 = hPv; Pvi = hPv; vi kPvk kvkThus kPvk kvk .For nite dimensional complex or real vector spaces, the standard inner product can be substituted for h; i .

Formulas A simple case occurs when the orthogonal projection is onto a line. If u is a unit vector on the line, thenthe projection is given by the outer product

Pu = uuT:

This operator leaves u invariant, and it annihilates all vectors orthogonal to u, proving that it is indeed the orthogonalprojection onto the line containing u.[3] A simple way to see this is to consider an arbitrary vector x as the sum of acomponent on the line (i.e. the projected vector we seek) and another perpendicular to it, x = xk + x? . Applyingprojection, we get

Pux = uuTxk + uuTx? = u(sign(uTxk)kxkk) + u ~0 = xk

by the properties of the dot product of parallel and perpendicular vectors.This formula can be generalized to orthogonal projections on a subspace of arbitrary dimension. Let u1, ..., uk bean orthonormal basis of the subspace U, and let A denote the n-by-k matrix whose columns are u1, ..., uk. Then theprojection is given by

PA = AAT [4]

which can be rewritten as

PA =Xi

hui; iui:

The matrix AT is the partial isometry that vanishes on the orthogonal complement of U and A is the isometry thatembeds U into the underlying vector space. The range of PA is therefore the nal space of A. It is also clear thatAAT is the identity operator on U.The orthonormality condition can also be dropped. If u1, ..., uk is a (not necessarily orthonormal) basis, and A is thematrix with these vectors as columns, then the projection is

PA = A(ATA)1AT: [5]

The matrix A still embeds U into the underlying vector space but is no longer an isometry in general. The matrix(ATA)1 is a normalizing factor that recovers the norm. For example, the rank-1 operator uuT is not a projection if||u|| 1. After dividing by uTu = ||u||2, we obtain the projection u(uTu)1uT onto the subspace spanned by u.When the range space of the projection is generated by a frame (i.e. the number of generators is greater than itsdimension), the formula for the projection takes the form: PA = A(ATA)+AT . Here A+ stands for the MoorePenrose pseudoinverse. This is just one of many ways to construct the projection operator.

36 CHAPTER 9. PROJECTION (LINEAR ALGEBRA)

If a matrix [A B] is non-singular and AT B = 0 (i.e., B is the null space matrix of A),[6] the following holds:

I = [A B][A B]1AT

BT

1AT

BT

= [A B]

AT

BT

[A B]

1 AT

BT

= [A B]

ATA OO BTB

1AT

BT

= A(ATA)1AT +B(BTB)1BT

If the orthogonal condition is enhanced to AT W B = AT WT B = 0 withW being non-singular, the following holds:

I =A B

(ATWA)1AT(BTWB)1BT

W:

All these formulas also hold for complex inner product spaces, provided that the conjugate transpose is used insteadof the transpose.

9.2.2 Oblique projectionsThe term oblique projections is sometimes used to refer to non-orthogonal projections. These projections are alsoused to represent spatial gures in two-dimensional drawings (see oblique projection), though not as frequently asorthogonal projections.Oblique projections are dened by their range and null space. A formula for the matrix representing the projectionwith a given range and null space can be found as follows. Let the vectors u1, ..., uk form a basis for the range of theprojection, and assemble these vectors in the n-by-kmatrixA. The range and the null space are complementary spaces,so the null space has dimension n k. It follows that the orthogonal complement of the null space has dimensionk. Let v1, ..., vk form a basis for the orthogonal complement of the null space of the projection, and assemble thesevectors in the matrix B. Then the projection is dened by

P = A(BTA)1BT:

This expression generalizes the formula for orthogonal projections given above.[7]

9.3 Canonical formsAny projection P = P2 on a vector space of dimension d over a eld is a diagonalizable matrix, since its minimalpolynomial is x2 x, which splits into distinct linear factors. Thus there exists a basis in which P has the form

P = Ir 0drwhere r is the rank of P. Here Ir is the identity matrix of size r, and 0dr is the zero matrix of size d r. If the vectorspace is complex and equipped with an inner product, then there is an orthonormal basis in which the matrix of Pis[8]

P =

1 10 0

1 k0 0

Im 0s

where 1 2 ... k > 0. The integers k, s, m and the real numbers i are uniquely determined. Note that 2k+ s + m = d. The factor Im 0s corresponds to the maximal invariant subspace on which P acts as an orthogonalprojection (so that P itself is orthogonal if and only if k = 0) and the i-blocks correspond to the oblique components.

9.4. PROJECTIONS ON NORMED VECTOR SPACES 37

9.4 Projections on normed vector spacesWhen the underlying vector space X is a (not necessarily nite-dimensional) normed vector space, analytic questions,irrelevant in the nite-dimensional case, need to be considered. Assume now X is a Banach space.Many of the algebraic notions discussed above survive the passage to this context. A given direct sum decompositionof X into complementary subspaces still species a projection, and vice versa. If X is the direct sum X = U V,then the operator dened by P(u + v) = u is still a projection with range U and kernel V. It is also clear that P2 = P.Conversely, if P is projection on X, i.e. P2 = P, then it is easily veried that (I P)2 = (I P). In other words, (I P) is also a projection. The relation I = P + (I P) implies X is the direct sum Ran(P) Ran(I P).However, in contrast to the nite-dimensional case, projections need not be continuous in general. If a subspace Uof X is not closed in the norm topology, then projection onto U is not continuous. In other words, the range of acontinuous projection P must be a closed subspace. Furthermore, the kernel of a continuous projection (in fact, acontinuous linear operator in general) is closed. Thus a continuous projection P gives a decomposition of X into twocomplementary closed subspaces: X = ran(P) ker(P) = ker(I P) ker(P).The converse holds also, with an additional assumption. Suppose U is a closed subspace of X. If there exists a closedsubspace V such that X = U V, then the projection P with range U and kernel V is continuous. This follows fromthe closed graph theorem. Suppose xn x and Pxn y. One needs to show that Px = y. Since U is closed and{Pxn} U, y lies in U, i.e. Py = y. Also, xn Pxn = (I P)xn x y. Because V is closed and {(I P)xn} V,we have x y V, i.e. P(x y) = Px Py = Px y = 0, which proves the claim.The above argument makes use of the assumption that bothU andV are closed. In general, given a closed subspaceU,there need not exist a complementary closed subspaceV, although for Hilbert spaces this can always be done by takingthe orthogonal complement. For Banach spaces, a one-dimensional subspace always has a closed complementarysubspace. This is an immediate consequence of HahnBanach theorem. Let U be the linear span of u. By HahnBanach, there exists a bounded linear functional such that (u) = 1. The operator P(x) = (x)u satises P2 =P, i.e. it is a projection. Boundedness of implies continuity of P and therefore ker(P) = ran(I P) is a closedcomplementary subspace of U.

9.5 Applications and further considerationsProjections (orthogonal and otherwise) play a major role in algorithms for certain linear algebra problems:

QR decomposition (see Householder transformation and GramSchmidt decomposition);

Singular value decomposition

Reduction to Hessenberg form (the rst step in many eigenvalue algorithms).

Linear regression

As stated above, projections are a special case of idempotents. Analytically, orthogonal projections are non-commutativegeneralizations of characteristic functions. Idempotents are used in classifying, for instance, semisimple algebras,while measure theory begins with considering characteristic functions of measurable sets. Therefore, as one canimagine, projections are very often encountered in the context operator algebras. In particular, a von Neumannalgebra is generated by its complete lattice of projections.

9.6 GeneralizationsMore generally, given a map between normed vector spaces T : V ! W; one can analogously ask for this map tobe an isometry on the orthogonal complement of the kernel: that (kerT )? ! W be an isometry (compare Partialisometry); in particular it must be onto. The case of an orthogonal projection is when W is a subspace of V. InRiemannian geometry, this is used in the denition of a Riemannian submersion.

38 CHAPTER 9. PROJECTION (LINEAR ALGEBRA)

9.7 See also Centering matrix, which is an example of a projection matrix. Orthogonalization Invariant subspace Properties of trace Dykstras projection algorithm to compute the projection onto an intersection of sets

9.8 Notes[1] Meyer, pp 386+387

[2] Meyer, p. 433

[3] Meyer, p. 431

[4] Meyer, equation (5.13.4)

[5] Meyer, equation (5.13.3)

[6] See also Linear least squares (mathematics) Properties of the least-squares estimators.

[7] Meyer, equation (7.10.39)

[8] Dokovi, D. . (August 1991). Unitary similarity of projectors. AequationesMathematicae 42 (1): 220224. doi:10.1007/BF01818492.

9.9 References Dunford, N.; Schwartz, J. T. (1958). Linear Operators, Part I: General Theory. Interscience. Meyer, Carl D. (2000). Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Math-ematics. ISBN 978-0-89871-454-8.

9.10 External links MIT Linear Algebra Lecture on Projection Matrices at Google Video, from MIT OpenCourseWare Planar Geometric Projections Tutorial a simple-to-follow tutorial explaining the dierent types of planargeometric projections.

Chapter 10

Projection-valued measure

In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function dened oncertain subsets of a xed set and whose values are self-adjoint projections on a xed Hilbert space. Projection-valuedmeasures are formally similar to real-valued measures, except that their values are self-adjoint projections rather thanreal numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respectto a PVM; the result of such an integration is a linear operator on the given Hilbert space.Projection-valued measures are used to express results in spectral theory, such as the important spectral theoremfor self-adjoint operators. The Borel functional calculus for self-adjoint operators is constructed using integrals withrespect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements. Theyare generalized by positive operator valued measures (POVMs) in the same sense that a mixed state or density matrixgeneralizes the notion of a pure state.

10.1 Formal denitionA projection-valued measure on a measurable space (X, M), where M is a -algebra of subsets of X, is a mapping from M to the set of self-adjoint projections on a Hilbert space H such that

(X) = idH

and for every , H, the set-function

E 7! h(E) j i

is a complex measure on M (that is, a complex-valued countably additive function). We denote this measure byS(; ) . Note that S(; ) is a real-valued measure, and a probability measure when has length one.If is a projection-valued measure and

E \ F = ;;

then (E), (F) are orthogonal projections. From this follows that in general,

(E)(F ) = (E \ F ) = (F )(E);

and they commute.Example. Suppose (X,M, ) is a measure space. Let (E) be the operator of multiplication by the indicator function1E on L2(X). Then is a projection-valued measure.

39

40 CHAPTER 10. PROJECTION-VALUED MEASURE

10.2 Extensions of projection-valued measures, integrals and the spectraltheorem

If is a projection-valued measure on (X, M), then the map

1E 7! (E)extends to a linear map on the vector space of step functions on X. In fact, it is easy to check that this map is a ringhomomorphism. This map extends in a canonical way to all bounded complex-valued measurable functions on X,and we have the following.Theorem. For any bounded M-measurable function f on X, there is a unique bounded linear operator T(f) suchthat

hT(f) j i =ZX

f(x)d S(; )(x)

for all , H. Here, S(; ) denotes the complex measure E 7! h(E) j i from the denition of . The map

f 7! T(f)is a homomorphism of rings. An integral notation is often used for T(f) , as in

T(f) =ZX

f(x)d(x) =

ZX

fd:

The theorem is also correct for unbounded measurable functions f, but then T(f) will be an unbounded linearoperator on the Hilbert space H.The spectral theorem says that every self-adjoint operator A : H ! H has an associated projection-valued measureA dened on the real axis, such that

A =

ZRxdA(x):

This allows to dene the Borel functional calculus for such operators: if g : R! C is a measurable function, we set

g(A) :=

ZRg(x)dA(x):

10.3 Structure of projection-valued measuresFirst we provide a general example of projection-valued measure based on direct integrals. Suppose (X, M, ) is ameasure space and let {Hx}x X be a -measurable family of separable Hilbert spaces. For every E M, let (E)be the operator of multiplication by 1E on the Hilbert space

Z X

Hx d(x):

Then is a projection-valued measure on (X, M).Suppose , are projection-valued measures on (X, M) with values in the projections of H, K. , are unitarilyequivalent if and only if there is a unitary operator U:H K such that

10.4. APPLICATION IN QUANTUM MECHANICS 41

(E) = U(E)U

for every E M.Theorem. If (X,M) is a standard Borel space, then for every projection-valued measure on (X,M) taking values inthe projections of a separable Hilbert space, there is a Borel measure and a -measurable family of Hilbert spaces{Hx}x X , such that is unitarily equivalent to multiplication by 1E on the Hilbert space

Z X

Hx d(x):

The measure class of and the measure equivalence class of the multiplicity function x dim Hx completely char-acterize the projection-valued measure up to unitary equivalence.A projection-valued measure is homogeneous of multiplicity n if and only if the multiplicity function has constantvalue n. Clearly,Theorem. Any projection-valued measure taking values in the projections of a separable Hilbert space is anorthogonal direct sum of homogeneous projection-valued measures:

=M

1n!(jHn)

where

Hn =

Z Xn

Hx d(jXn)(x)

and

Xn = fx 2 X : dimHx = ng:

10.4 Application in quantum mechanicsIn quantummechanics, the unit sphere of the Hilbert spaceH is interpreted as the set of possible states of a quantumsystem, the measurable space X is the value space for some quantum property of the system (an observable), andthe projection-valued measure expresses the probability that the observable takes on various values.A common choice for X is the real numbers, but it may also be R3 (for position or momentum), a discrete set (forangular momentum, energy of a bound state, etc), or the 2-point set true and false for the truth-value of anarbitrary proposition about .Let E be a measurable subset of X and a state in H, so that ||=1. The probability that the observable takes itsvalue in E given the system in state is

P = h; (E)()i = hj(E)ji;where the latter notation is preferred in physics. We can parse this in two ways. First, for each xed E, the projection(E) is a self-adjoint operator on H whose 1-eigenspace is the states for which the value of the observable alwayslies in E, and whose 0-eigenspace is the states for which the value of the observable never lies in E. Second, foreach xed , the association E ,() is a probability measure on X making the values of the observable intoa random variable.A measurement that can be performed by a projection-valued measure is called a projective measurement. If Xis the real numbers, there is associated to a Hermitian operator A dened on H by

42 CHAPTER 10. PROJECTION-VALUED MEASURE

A() =

ZR d()();

which takes the more readable form

A() =Xi

i(i)()

if the support of is a discrete subset of R. This operator is called an observable in quantum mechanics.

10.5 GeneralizationsThe idea of a projection-valued measure is generalized by the positive operator-valued measure (POVM), where theneed for the orthogonality implied by projection operators is replaced by the idea of a set of operators that are anon-orthogonal partition of unity. This generalization is motivated by applications to quantum information theory.

10.6 References G. W. Mackey, The Theory of Unitary Group Representations, The University of Chicago Press, 1976 M. Reed and B. Simon, Methods of Mathematical Physics, vols IIV, Academic Press 1972. G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrdinger Operators, http://www.mat.univie.ac.at/~{}gerald/ftp/book-schroe/, American Mathematical Society, 2009.

V. S. Varadarajan, Geometry of Quantum Theory V2, Springer Verlag, 1970.

Chapter 11

Projectivization

In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective spaceP(V ) , whose elements are one-dimensional subspaces of V. More generally, any subset S of V closed under scalarmultiplication denes a subset of P(V ) formed by the lines contained in S and is called the projectivization of S.

11.1 Properties Projectivization is a special case of the factorization by a group action: the projective space P(V ) is the quotientof the open set V\{0} of nonzero vectors by the action of the multiplicative group of the base eld by scalartransformations. The dimension of P(V ) in the sense of algebraic geometry is one less than t