EIGENVALUES, EIGENVECTORS, AND...

8
CHAPTER 5 EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION Note: In these definitions v R n and λ R, but sometimes it is necessary to extend the domain of T to allow v C n and λ C. Example: reflection operator T about the line y = (1/2)x b 1 is an eigenvector of T corresponding to the eigenvalue 1. b 2 is an eigenvector of T corresponding to the eigenvalue -1. Definition x y L b 2 b 1 = T (b 1 ) T (b 2 )= -b 2

Transcript of EIGENVALUES, EIGENVECTORS, AND...

Page 1: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATIONcase.ntu.edu.tw/CASTUDIO/Files/speech/Ref/LA5.1.pdf · EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION Note: In these definitions v ∈

CHAPTER 5 EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION

Note: In these definitions v ∈ Rn and λ ∈ R, but sometimes it is necessary to extend the domain of T to allow v ∈ Cn and λ ∈ C.

Example: reflection operator T about the line y = (1/2)x

b1 is an eigenvector of T corresponding to the eigenvalue 1.

b2 is an eigenvector of T corresponding to the eigenvalue -1.

Definition

x

y

Lb2

b1 = T (b1)

T (b2) = �b2

Page 2: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATIONcase.ntu.edu.tw/CASTUDIO/Files/speech/Ref/LA5.1.pdf · EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION Note: In these definitions v ∈

Note: In these definitions v ∈ Rn and λ ∈ R, but sometimes it is necessary to allow v ∈ Cn and λ ∈ C.

Example:

non-eigenvectors eigenvectors

Definition

A =

0.6 0.80.8 �0.6

A

�55

�=

�, A

76

�=

�, A

21

�=

�, A

�12

�=

Page 3: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATIONcase.ntu.edu.tw/CASTUDIO/Files/speech/Ref/LA5.1.pdf · EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION Note: In these definitions v ∈

Example:

An eigenvector of A corresponds to a unique eigenvalue. An eigenvalue of A has infinitely many eigenvectors.

A =

2

45 2 1�2 1 �12 2 4

3

5 v =

2

41�11

3

5

Proof T(v) = λv ⇔ Av = λv.

The eigenvectors and corresponding eigenvalues of a linear operator are the same

as those of its standard matrix.

Property

Av =

2

45 2 1�2 1 �12 2 4

3

5

2

41�11

3

5 =

2

44�44

3

5 = 4

2

41�11

3

5 = 4v

A(cv) = c(Av) = c(4v) = 4(cv), 8c 2 R

Page 4: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATIONcase.ntu.edu.tw/CASTUDIO/Files/speech/Ref/LA5.1.pdf · EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION Note: In these definitions v ∈

Proof Av = λv ⇔ Av - λv = 0 ⇔ Av - λInv = 0 ⇔ (A - λIn)v = 0.

Example: to check 3 and -2 are eigenvalues of the linear operator

Definition

Definition

T

✓x1

x2

�◆=

�2x2

�3x1 + x2

Property Let A be an n⇥nmatrix with eigenvalue �. The eigenvectors of A corresponding

to � are the nonzero solutions of (A� �In)v = 0.

Page 5: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATIONcase.ntu.edu.tw/CASTUDIO/Files/speech/Ref/LA5.1.pdf · EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION Note: In these definitions v ∈

consider its standard matrix

The rank of A - 3I2 is 1, so its null space is not the zero space, and every nonzero vector in the null space is an eigenvector of T corresponding to the eigenvalue 3.

Similarly, the rank of A + 2I2 is 1, so its null space is not the zero space, and every nonzero vector in the null space is an eigenvector of T corresponding to the eigenvalue -2.

A =

0 �2�3 1

Page 6: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATIONcase.ntu.edu.tw/CASTUDIO/Files/speech/Ref/LA5.1.pdf · EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION Note: In these definitions v ∈

Example: to check that 3 is an eigenvalue of B and find a basis for the corresponding eigenspace, where

find the reduced row echelon form of B - 3I3 and the solution set of (B - 3I3)x = 0, respectively, to be

Thus {[ 1 0 0 ]T, [ 0 1 1 ]T} is a basis of the eigenspace of B corresponding to the eigenvalue 3.

Example: some square matrices and linear operators on Rn have no

real eigenvalues, like the 90º-rotation matrix

B =

2

43 0 00 1 20 2 1

3

5

2

40 1 �10 0 00 0 0

3

5

2

4x1

x2

x3

3

5 =

2

4x1

x3

x3

3

5 = x1

2

4100

3

5+ x3

2

4011

3

5

0 �11 0

Page 7: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATIONcase.ntu.edu.tw/CASTUDIO/Files/speech/Ref/LA5.1.pdf · EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION Note: In these definitions v ∈

7

T (x) = Ax = �xAn eig. vec. x ===> unique eig. val. λ.

An eig. val. λ ===> multiple eig. vectors x. ===> The set of all eig. vectors corr. to λ is {x 2 Rn : (A� �In)x = 0} \ {0}

eigenspace corr. to the eig. val. λ

Page 8: EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATIONcase.ntu.edu.tw/CASTUDIO/Files/speech/Ref/LA5.1.pdf · EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION Note: In these definitions v ∈

Section 5.1: Problems 1, 3, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59

Homework Set for Sections 5.1