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Linear Algebra. Lecture 36. Revision Lecture I. Seg V and III. Eigenvalues and Eigenvectors. 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 . …. - PowerPoint PPT Presentation

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Page 1: Linear Algebra
Page 2: Linear Algebra
Page 3: Linear Algebra
Page 4: 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

Page 5: Linear Algebra

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

Page 6: Linear Algebra

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

Page 7: Linear Algebra

det( ) 0A I

Page 8: Linear Algebra

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.

Page 9: Linear Algebra

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

Page 10: Linear Algebra

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.

Page 11: Linear Algebra

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

Page 12: Linear Algebra

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

Page 13: Linear Algebra
Page 14: Linear Algebra

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 …

Page 15: Linear Algebra

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

Page 16: Linear Algebra
Page 17: Linear Algebra

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

Page 18: Linear Algebra

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

Page 19: Linear Algebra

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

Page 20: Linear Algebra
Page 21: Linear Algebra

Similarity of two matrix representations: A=PCP-1

Page 22: Linear Algebra
Page 23: Linear Algebra

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

Page 24: Linear Algebra

x xr r

rB r Bx xB B

Page 25: Linear Algebra
Page 26: Linear Algebra

1 kAx kx

Page 27: Linear Algebra

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.

Page 28: Linear Algebra

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.

Page 29: Linear Algebra
Page 30: Linear Algebra

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

Page 31: Linear Algebra

(0)

0

x Axx x

SolveSubject to

Page 32: Linear Algebra

( ) tt ve

x Ax

x

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

Page 33: Linear Algebra

( )

( )

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…

Page 34: Linear Algebra

( ) 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.

Page 35: Linear Algebra
Page 36: Linear Algebra

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

Page 37: Linear Algebra

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

Page 38: Linear Algebra

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.

Page 39: Linear Algebra

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

of entry aij

(or the ijth cofactor of A).

Page 40: Linear Algebra

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

Page 41: Linear Algebra

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

Page 42: Linear Algebra

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

Page 43: Linear Algebra

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

Page 44: Linear Algebra
Page 45: Linear Algebra

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.

…..

Page 46: Linear Algebra

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.

Page 47: Linear Algebra

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)

Page 48: Linear Algebra
Page 49: Linear Algebra

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

Page 50: Linear Algebra

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

Page 51: Linear Algebra

Let A be an invertible matrix, then

1 1det

A adj AA

Page 52: Linear Algebra

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}

Page 53: Linear Algebra

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}

Page 54: Linear Algebra