linear algebra

If A is an n x n matrix, then a scalar is called an eigenvalue of A if there is a nonzero vector x such that Ax = x.

…

λ

λ

If A is a triangular matrix then the eigenvalues of A are the entries on the main diagonal of A.

If is an eigenvalue of a matrix A and x is a corresponding eigenvector, and if k is any positive integer, then is an eigenvalue of Ak and x is a corresponding eigenvector.

λ

kλ

det( ) 0A I

If A and B are n x n matrices, then A is similar to B if there is an invertible matrix P such that P -1AP = B, or equivalently, A = PBP -1.

…

If n x n matrices A and B are similar, then they have the same characteristic polynomial and hence the same Eigenvalues.

A square matrix A is said to be diagonalizable if A is similar to a diagonal matrix.

i.e. if A = PDP -1 for some invertible matrix P and some diagonal matrix D.

An n x n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors.

An n x n matrix with n distinct eigenvalues is diagonalizable.

Let V and W be n-dim and m-dim spaces, and T be a LT from V to W. To associate a matrix with T we chose bases B and C for V and W respectively …

Given any x in V, the coordinate vector [x]B is in Rn and the [T(x)]C coordinate vector of its image, is in Rm

…

Let {b1 ,…,bn} be the basis B for V.

If x = r1b1 +…+ rnbn, then

1

[ ]Bx

n

r

r

1

1

( ) ( )

( ) ( )n

n

r r

r r

and

1 n

1 n

T x T b b

T b T b …

Connection between [ x ]B and [T(x)]C

1[ ( )] [ ( )] [ ( )]C 1 C n CT x T b T b nr r

[ ( )] [ ]C BT x M x

[ ( )] [ ( )] [ ( )]

where1 C 2 C n CM T b T b T b

This equation can be written as

The Matrix M is the matrix representation of T, Called the matrix for T relative to the bases B and C

Similarity of two matrix representations: A=PCP-1

A complex scalar satisfies

if and only if there is a nonzero vector x in Cn such that We call a (complex) eigenvalue and x a (complex) eigenvector corresponding to .

λdet( - ) 0A I

λ

λ

.Ax x

x xr r

rB r Bx xB B

1 kAx kx

If A has two complex eigenvalues whose absolute value is greater than 1, then 0 is a repellor and iterates of x0 will spiral outward around the origin.

…

If the absolute values of the complex eigenvalues are less than 1, the origin is an attractor and the iterates of x0 spiral inward toward the origin.

x Ax1 1

11 1

1

( ) ( )( ) , ( ) ,

( ) ( )x x

A

n n

n

n nn

x t x tt t

x t x t

a a

a a

where

and

(0)

0

x Axx x

SolveSubject to

( ) tt ve

x Ax

x

For the generalequationSolution might be a linear combination of the form …

( )

( )

t

t

t e

t e

x v

Ax Av0,

( )

tet (t)

Since

iff ,i.e. iff λ is aneigen value ofand is a corresponding eigenvector.

x Ax v AvA

v…

( ) tt e of .x v x Ax

Thus each eigenvalue - eigenvector pair provides a solution Such solutions are sometimes called eigen functions of the differential equation.

3 x 3 Determinant

11 12 13

21 22 23

31 32 33

11 22 33 12 23 31 13 21 32

13 22 31 11 23 32 12 21 33

det( )a a a

A a a aa a a

a a a a a a a a aa a a a a a a a a

11 12 13

21 22 23

31 32 33

a a aA a a a

a a a

11 11 12 12

11 1

det det det

... ( 1) detnn n

A a A a A

a A

Expansion

11 1

1

( 1) detn

jj j

j

a A

Minor of a MatrixIf A is a square matrix, then the Minor of entry aij (called the ijth minor of A) is denoted by Mij and is defined to be the determinant of the sub matrix that remains when the ith row and jth column of A are deleted.

CofactorThe number Cij=(-1)i+j Mij is called the cofactor

of entry aij

(or the ijth cofactor of A).

Cofactor Expansion Across the First Row

11 11 12 12 1 1det ... n nA a C a C a C

( 1) deti ji j ijC A

The determinant of a matrix A can be computed by a cofactor expansion across any row or down any column.

The cofactor expansion across the ith row

1 1 2 2det ...i i i i in inA a C a C a C

The cofactor expansion down the jth column

1 1 2 2det ...j j j j nj njA a C a C a C

If A is triangular matrix, then det (A) is the product of the entries on the main diagonal.

11

21 22

31 32 33

41 42 43 44

0 0 00 0

0

aa a

Aa a aa a a a

11 22 33 44det( )A a a a a

Let A be a square matrix.

If a multiple of one row of A is added to another row to produce a matrix B, then det B = det A.

…..

ContinueIf two rows of A are

interchanged to produce B, then det B = –det A.If one row of A is multiplied

by k to produce B, then det B = k det A.

If A is an n x n matrix, thendet AT = det A.

If A and B are n x n matrices, then

det (AB)=(det A )(det B)

ObserveFor any n x n matrix A and any b in Rn, let Ai(b) be the matrix obtained from A by replacing column i by the vector b.

1( ) ... ...i nA b a b a

coli

Let A be an invertible n x n matrix. For any b in Rn, the unique solution x of Ax = b has entries given by

det ( ), 1,2,...,

deti

iA b

x i nA

Let A be an invertible matrix, then

1 1det

A adj AA

Let T: R2 R2 be the linear transformation determined by a 2 x 2 matrix A. If S is a parallelogram in R2, then{area of T (S)} = |detA|. {area of S}

If T is determined by a 3 x 3 matrix A, and if S is a parallelepiped in R3, then{volume of T (S)} = |detA|. {volume of S}