book1, chapter 6: matrix, author: n.d. ingham. pheniox publications

8/13/2019 Book1, Chapter 6: matrix, Author: N.D. ingham. Pheniox publications.

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Chapter 6

Eigenvalues and Eigenvectors


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6.1 DefinitionsDefinition 1:A nonzero vector xis an eigenvector(or characteristic vector)

of a square matrix Aif there exists a scalar

such that Ax =

x. Then

is aneigenvalue(or characteristic value) of A.

Note: The zero vector can not be an eigenvector even though A0 = 0. But

= 0 can be an eigenvalue.

Example:

Show x 21 isan eigenvector for A 2 4

3 6

Solution:Ax2 4

3 6

2

1

0

0

But for 0, x 02

1

0

0

Thus,x is an eigenvectorof A,and 0isaneigenvalue.


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Geometric interpretation ofEigenvalues and Eigenvectors

An nn matrix Amultiplied by n1 vector xresults in another

n1 vector y=Ax. Thus Acan be considered as atransformation matrix.

In general, a matrix acts on a vector by changing both its

magnitude and its direction. However, a matrix may act oncertain vectors by changing only their magnitude, and leaving

their direction unchanged (or possibly reversing it). These

vectors are the eigenvectorsof the matrix.

A matrix acts on an eigenvector by multiplying its magnitude bya factor, which is positive if its direction is unchanged and

negative if its direction is reversed. This factor is the eigenvalue

associated with that eigenvector.


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6.2 EigenvaluesLet x be an eigenvector of the matrix A.Then there must exist an

eigenvalue such that Ax = x or, equivalently,

Ax - x= 0 or

(AI)x= 0

If we define a new matrix B = AI, then

Bx= 0

If Bhas an inverse then x = B-10 = 0. But an eigenvector cannot

be zero.

Thus, it follows that x will be an eigenvector of Aif and only if Bdoes not have an inverse, or equivalently det(B)=0, or

det(AI)= 0

This is called the characteristic equation of A. Its roots

determine the eigenvalues of A.


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Example 1: Find the eigenvalues of

two eigenvalues: 1, 2Note:The roots of the characteristic equation can be repeated. That is, 1= 2

== k. If that happens, the eigenvalue is said to be of multiplicity k.

Example 2: Find the eigenvalues of

= 2 is an eigenvector of multiplicity 3.

51

122A

)2)(1(23

12)5)(2(51

122

2

AI

6.2 Eigenvalues: examples

200

020

012

A

0)2(

200

020

0123

AI


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Example 1 (cont.):

00

41

41

123)1(:1 AI

0,1

4

,404

2

1

1

2121

ttx

x

txtxxx

x

00

3131124)2(:2 AI

0,1

3

2

1

2

ss

x

xx

6.3 EigenvectorsTo each distinct eigenvalue of a matrix Athere will correspond at least one

eigenvector which can be found by solving the appropriate set of homogenous

equations. If iis an eigenvalue then the corresponding eigenvector xiis the

solution of (AiI)xi= 0


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Example 2 (cont.): Find the eigenvectors of

Recall that = 2 is an eigenvector of multiplicity 3.

Solve the homogeneous linear system represented by

Let . The eigenvectors of = 2 are of the

form

sand tnot both zero.

00

0

000000

010

)2(

3

2

1

x

x

x

AI x

txsx 31 ,

,

1

00

0

01

0

3

2

1

ts

t

s

x

xx

x

6.3 Eigenvectors

200

020

012

A


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6.4 Properties of Eigenvalues and Eigenvectors

Definition:The trace of a matrix A, designated by tr(A), is the sum

of the elements on the main diagonal.

Property 1:The sum of the eigenvalues of a matrix equals the

trace of the matrix.

Property 2:A matrix is singular if and only if it has a zeroeigenvalue.

Property 3:The eigenvalues of an upper (or lower) triangularmatrix are the elements on the main diagonal.

Property 4:If is an eigenvalue of Aand Ais invertible, then 1/

is an eigenvalue of matrix A-1.


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6.4 Properties of Eigenvalues andEigenvectors

Property 5:If is an eigenvalue of A then kis an eigenvalue ofkAwhere kis any arbitrary scalar.

Property 6:Ifis an eigenvalue of A then

k

is an eigenvalue ofAkfor any positive integer k.

Property 8:If is an eigenvalue of A then is an eigenvalue of AT.

Property 9:The product of the eigenvalues (counting multiplicity)

of a matrix equals the determinant of the matrix.


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6.5 Linearly independent eigenvectors

Theorem:Eigenvectors corresponding to distinct (that is, different)

eigenvalues are linearly independent.Theorem:If is an eigenvalue of multiplicity kof an nnmatrix

A then the number of linearly independent eigenvectors of A

associated with is given by m = n - r(A- I). Furthermore, 1 m

k.

Example 2 (cont.): The eigenvectors of = 2 are of the form

sand tnot both zero.

= 2 has two l inear ly independent eigenvectors

,10

0

00

1

0

3

2

1

tst

s

x

x

x

x

book1, chapter 6: matrix, author: n.d. ingham. pheniox publications

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