boot camp in linear algebra joel barajas karla l caballero university of california silicon valley...
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Boot Camp in Linear Algebra
Joel BarajasKarla L CaballeroUniversity of California
Silicon Valley Center
October 8th, 2008.
Matrices A matrix is a rectangular array of numbers
(also called scalars), written between square brackets, as in
Vectors A vector is defined as a matrix with only
one column or row
Row vectorColumn vector or vector
Zero and identity matrices
The zero matrix (of size m X n) is the matrix with all entries equal to zero
An identity matrix is always square and its diagonal entries are all equal to one, otherwise are zero. Identity matrices are denoted by the letter I.
Vector Operations The inner product (a.k.a. dot product or
scalar product) of two vectors is defined by
The magnitude of a vector is
Vector Operations The projection of vector
y onto vector x is
where vector ux has unit magnitude and the same direction as x
Vector Operations The angle between vectors x and y is
Two vectors x and y are said to be orthogonal if xTy=0 orthonormal if xTy=0 and |x|=|y|=1
Vector Operations A set of vectors x1, x2, …, xn are said to be linearly
dependent if there exists a set of coefficients a1, a2, …, an (at least one different than zero) such that
A set of vectors x1, x2, …, xn are said to be linearly independent if
Matrix OperationsMatrix transpose If A is an m X n matrix, its transpose,
denoted AT, is the n X m matrix given by (AT )ij = Aji. For example,
Matrix OperationsMatrix addition Two matrices of the same size can be
added together, to form another matrix (of the same size), by adding the corresponding entries
Matrix OperationsScalar multiplication The multiplication of a matrix by a scalar
(i.e., number), is done by multiplying every entry of the matrix by the scalar
Matrix OperationsMatrix multiplication You can multiply two matrices A and B
provided their dimensions are compatible, which means the number of columns of A equals the number of rows of B. Suppose that A has size m X p and B has size p X n. The product matrix C = AB, which has size m X n, is defined by
Matrix Operations The trace of a square matrix Ad×d is the sum of its
diagonal elements
The rank of a matrix is the number of linearly
independent rows (or columns)
A square matrix is said to be non-singular if and only if its rank equals the number of rows
(or columns) A non-singular matrix has a non-zero determinant
Matrix Operations A square matrix is said to be orthonormal
if AAT=ATA=I For a square matrix A
if xTAx>0 for all x≠0, then A is said to be positive-definite (i.e., the covariance matrix)
if xTAx≥0 for all x≠0, then A is said to be positive-semidefinite
Matrix inverse If A is square, and there is a matrix F such that FA
= I, then we say that A is invertible or nonsingular.
We call F the inverse of A, and denote it A-1. We can then also define A-k = (A-1)k. If a matrix is not invertible, we say it is singular or noninvertible.
A n×n A−1n× n= A
−1n×n A n×n= I
[a11 a12a21 a22 ]
−1
=1
a11a22−a21 a12 [ a22 −a12−a21 a11 ]
Matrix Operations The pseudo-inverse matrix A† is typically
used whenever A-1 does not exist (because A is not square or A is singular):
Matrix Operations The n-dimensional space in which all the n-
dimensional vectors reside is called a vector space
A set of vectors {u1, u2, ... un} is said to form a basis for a vector space if any arbitrary vector x can be represented by a linear combination of the {ui}
Matrix Operations The coefficients {a1, a2, ... an} are called the
components of vector x with respect to the basis {ui}
In order to form a basis, it is necessary and sufficient that the {ui} vectors are linearly independent
Matrix Operations A basis {ui} is said to be orthogonal if
A basis {ui} is said to be orthonormal if
Linear Transformations A linear transformation is a mapping from a vector space XN
onto a vector space YM, and is represented by a matrix Given vector x∈XN, the corresponding vector y on YM is
computed as
A linear transformation represented by a square matrix A is said to be orthonormal when AAT=ATA=I
Eigenvectors and Eigenvalues Let A be any square matrix. A scalar is
called and eigenvalue of A if there exists a non zero vector v such that:
Av=v
Any vector v satisfying this relation is called and eigenvector of A belonging to the eigenvalue of
How to compute the Eigenvalues and the Eigenvectors• Find the characteristic polynomial (t) of A.• Find the roots of (t) to obtain the
eigenvalues of A.• Repeat (a) and (b) for each eigenvalue of A.
a. Form the matrix M=A-I by subtracting down the diagonal A.
b. Find the basis for the solution space of the homogeneous system MX=0. (These basis vectors are linearly independent eigenvectors of A belonging to .)
Example We have a matrix
The characteristic polynomial (t) of A is computed. We have
13
24=A
)+)(t(=t=Δ(t)
==A
==tr(A)
25-t103t
1064
314
2
Example Set (t)=(t-5)(t+2)=0. The roots 1=5 and 2=-2
are the eigenvalues of A. We find an eigenvector v1 of A belonging to the
eigenvalue 1=5
)(=v
=
y
x==MX
==M
λIA=M
2,1
0
0
63
210
63
21
50
05
13
24
1