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5.1Orthogonality
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A set of vectors is called an orthogonal set if all pairs of distinct vectors in the set are orthogonal.
An orthonormal set is an orthogonal set of unit vectors.
An orthogonal (orthonormal) basis for a subspace W of
Rn is a basis for W that is an orthogonal (orthonormal)
set. An orthogonal matrix is a square matrix whose columns
form an orthonormal set.
Definitions
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1) Is the following set of vectors orthogonal? orthonormal?
2) Find an orthogonal basis and an orthonormal basis
for the subspace W of Rn
},...,,{ b)
2
1
1
,
1
4
2
,
2
1
3
a) 21 neee
Examples
02: W
zyx
z
y
x
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All vectors in an orthogonal set are linearly independent.
Let {v1, v2,…, vk } be an orthogonal basis for a subspace
W of Rn and w
be any vector in W. Then the unique
scalars c1 ,c2 , …, ck such that w = c1v1 + c2v2 + …+ ckvk
are given by
Theorems
kivv
vw
ii
ii ,...,1for c
Proof: To find ci we take the dot product with vi w vi = (c1v1 + c2v2 + …+ ckvk ) vi
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4) Is the following matrix orthogonal?
If it is orthogonal, find its inverse and its transpose.
cossin
sincos
010
001
100
B
212
141
123
A C
Examples3) The orthogonal basis for the subspace W in previous example is
Pick a vector in W and express it in terms of the vectorsin the basis.
1
1
1
,
0
1
1
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The following statements are equivalent for a matrix A :1) A is orthogonal 2) A
-1 = A
T
3) ||Av|| = ||v|| for every v in Rn
4) Av1∙ Av2 = v1∙ v2 for every v1 ,v2 in Rn
Theorems on Orthogonal Matrix
Let A be an orthogonal matrix. Then1) its rows form an orthonormal set. 2) A
-1 is also orthogonal.
3) |det(A)| = 14) |λ| = 1 where λ is an eigenvalue of A5) If A and B are orthogonal matrices, then so is AB
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5.2Orthogonal Complements
and Orthogonal Projections
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Recall: A normal vector n to a plane is orthogonal to every vector in that plane. If the plane passes through the origin, then it is a subspace W of R3 .
Also, span(n) is also a subspace of R3 Note that every vector in span(n) is orthogonal to
every vector in subspace W . Then span(n) is called orthogonal complement of W.
A vector v is said to be orthogonal to a subspace W
of Rn if it is orthogonal to all vectors in W.
The set of all vectors that are orthogonal to W is called the orthogonal complement of W, denoted W ┴ . That is
Orthogonal Complements
W} 0 :R{W wwvv n
Definition:
http://www.math.tamu.edu/~yvorobet/MATH304-2011C/Lect3-02web.pdf
W perp
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1) Find the orthogonal complements for W of R3 .
02: c)
1
1
0
and
0
1
1
vectorsby) spanned (subspace direction withplane b)
3
2
1
span a)
zyx
z
y
x
W
W
W
Example
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Let W be a subspace of Rn .
1) W ┴ is a subspace of Rn .
2) (W ┴)┴ = W3) W ∩ W ┴ = {0}4) If W = span(w1,w2,…,wk), then v is in W ┴ iff v∙wi = 0
for all i =1,…,k.
Theorems
Let A be an m x n matrix. Then(row(A))┴ = null(A) and (col(A))┴ = null(AT)
Proof?
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2) Use previous theorem to find the orthogonal complements
for W of R3 .
1 0
a) plane with direction (subspace spanned by) vectors 1 and 1
0 1
3 1
2 2
b) subspace spanned by vectors , an0 2
1 0
4 1
W
W
3
2
d 6
2
5
Example
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1 2v
v
u v u vproj v v
v vv
perp
2 1
w u
w = u u - w
Let u and v be nonzero vectors. w1 is called the vector component of u along v
(or projection of u onto v), and is denoted by projvu w2 is called the vector component of u orthogonal to v
w2 w1
u
v
Orthogonal Projections
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Let W be a subspace of Rn with an orthogonal basis
{u1, u2,…, uk }, the orthogonal projection of v onto W is defined as:
projW v = proju1 v + proju2 v + … + projuk v
The component of v orthogonal to W is the vectorperpW v = v – projw v
Orthogonal Projections
Let W be a subspace of Rn and v
be any vector in R
n .
Then there are unique vectors w1 in W and w2 in W ┴
such that v = w1 + w2 .
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3) Find the orthogonal projection of v = [ 1, -1, 2 ] onto W and the component of v orthogonal to W.
1
a) span 2
3
1 -1
b) subspace spanned by 1 and 1
0 1
c) : 2 0
W
W
x
W y x y z
z
Examples
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5.3The Gram-Schmidt Process
And the QR Factorization
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Goal: To construct an orthogonal (orthonormal) basis for
any subspace of Rn.
We start with any basis {x1, x2,…, xk }, and “orthogonalize” each vector vi in the basis one at a time by finding the component of vi orthogonal to W = span(x1, x2,…, xi-1 ).
The Gram-Schmidt Process
Let {x1, x2,…, xk } be a basis for a subspace W. Then choose the following vectors:
v1 = x1,v2 = x2 – projv1 x2
v3 = x3 – projv1 x3 – projv2 x3
… and so on Then {v1, v2,…, vk } is orthogonal basis for W . We can normalize each vector in the basis to form an
orthonormal basis.
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1) Use the following basis to find an orthonormal basis for R2
2) Find an orthogonal basis for R3 that contains the vector
,2
1,
1
3
Examples
1
2
1
1
0
1
,
1
1
1
,
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Note: Since Q is orthogonal, Q-1 = QT and we have R = QT A
The QR Factorization
If A is an m x n matrix with linearly independent columns, then A can be factored as A = QR where R is an invertible upper triangular matrix and Q is an m x n orthogonal matrix. In fact columns of Q form orthonormal
basis for Rn which can be constructed from columns of A
by using Gram-Schmidt process.
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3) Find a QR factorization for the following matrices.
11-1
012
1-1-1
A
21
13A
Examples
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5.4Orthogonal Diagonalization
of Symmetric Matrices
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1) Diagonalize the matrix.
62
23A
Example
Recall: A square matrix A is symmetric if AT = A. A square matrix A is diagonalizable if there exists a
matrix P and a diagonal matrix D such that P-1AP = D.
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A square matrix A is orthogonally diagonalizable if there exists an orthogonal matrix Q and a diagonal matrix D such that Q-1AQ = D.
Note that Q-1 = QT
Orthogonal Diagonalization
Definition:
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1. If A is orthogonally diagonalizable, then A is symmetric.2. If A is a real symmetric matrix, then the eigenvalues of A
are real.3. If A is a symmetric matrix, then any two eigenvectors
corresponding to distinct eigenvalues of A are orthogonal.
Theorems
A square matrix A is orthogonally diagonalizable if and only if it is symmetric.
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2) Orthogonally diagonalize the matrix
and write A in terms of matrices Q and D.
011
101
110
A
Example
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If A is orthogonally diagonalizable, and QTAQ = D then A can written as
where qi is the orthonormal column of Q, and λi is the corresponding eigenvalue.
A 222111T
nnnTT qq...qqqq
Theorem
This fact will help us construct the matrix A giveneigenvalues and orthogonal eigenvectors.
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3) Find a 2 x 2 matrix that has eigenvalues 2 and 7, withcorresponding eigenvectors
2
1 v
1
2 v 21
Example
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5.5Applications
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A quadratic form in x and y :
A quadratic form in x,y and z:
Quadratic Forms
2 2 2ax by cz dxy exz fyz
2 2ax by cxy 12
12
T a c
c b
x x
1 12 2
1 12 2
1 12 2
T
a d e
d b f
e f c
x x
where x is the variable (column) matrix.
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A quadratic form in n variables is a function
f : Rn R of the form:
where A is a symmetric n x n matrix and x is in Rn
Quadratic Forms
( ) Tf Ax x x
A is called the matrix associated with f.
2 2
2 2
( , ) 8
( , ) 2 5
z f x y x y xy
z f x y x y
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The Principal Axes Theorem
Example: Find a change of variable that transforms theQuadratic into one with no cross-product terms.
Every quadratic form can be diagonalized. In fact,if A is a symmetric n x n matrix and if Q is an orthogonal matrix so that QTAQ = D then the change of variable x = Qy transforms the quadratic form into 2 2 2
1 1 2 2 A T Tn nD y y ... y x x y y
2 2
2 2
( , ) 8
( , ) 2 5
z f x y x y xy
z f x y x y