Determinants and eigenvalues

[Pages:8]Determinants and eigenvalues

Math 40, Introduction to Linear Algebra Wednesday, February 15, 2012

Amazing facts about determinants

det A can be found by "expanding" along any row or any column

Consequence:

Theorem. The determinant of a triangular matrix is the product of its diagonal entries.

123 4

A = 00

5 0

6 8

7 9

0 0 0 10

det(A) = 1 ? 5 ? 8 ? 10 = 400

Amazing facts about determinants

EROs barely change the determinant, and they do so in a predictable way.

EROs

swap two rows

multiply row by scalar c add c row i

to row j

effect on det A

changes sign multiply det by scalar c

no change at all!

=

Strategy to compute det A more quickly for general matrices A

Perform EROs to get REF of A and

compute det A based on det of REF

Amazing facts about determinants

Theorem. A square matrix A is invertible if and only if det A = 0.

det(A) = det(AT )

determinant is multiplicative

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

Consequence:

det(A) det(A-1) = det(AA-1)

= det(I) = 1

If

A

is

invertible,

det(A-1)

=

1 det

A

.

det(A-1)

=

1 det A

Example using properties of determinant

Example If det A = -3 for a 5 x 5 matrix A, find the determinant of the matrix 4A3.

We have

det(4A3) = 45 det(A3) = 45[det(A)]3 = 45(-3)3

Another property of the determinant?

Question True or false: det(A + B) = det A + det B ?

Consider

False!

A=

1 1

1 1

,

B=

1 -1

-1 1

Then

A

+

B

=

2 0

0 2

det(A + B) = 4 = 0 = det A + det B

Eigenvalues and eigenvectors

Introduction to eigenvalues

Let A be an n x n matrix.

If Ax = x for some scalar

and some nonzero vector x , then we say is an eigenvalue of A and x is an eigenvector associated with .

Viewed as a linear transformation from Rn to Rn A sends vector x to a scalar multiple of itself (x ).

Eigenvalues, eigenvectors for a 2x2 matrix

A=

1 5

2 4

1 2 2 12

2

Notice that

5 4 5 = 30 = 6 5

12 54

6 15

=6

6 15

eigenvectors

eigenvalues

12 54

-1 1

1 = -1

= (-1)

-1 1

6

2

15 = 3 5

Any (nonzero) scalar multiple of an eigenvector is itself an eigenvector (associated w/same eigenvalue).

A(cx) = c(Ax) = c(x) = (cx)

Graphic demonstration of eigenvalues and eigenvectors: eigshow

eigshow demonstrates how the image Ax changes as we rotate a unit vector x in R2 around a circle

in particular, we are interested in

knowing when Ax is parallel to x

Finding eigenvalues of A

We want nontrivial solutions to Ax = x Ax - x = 0 Ax - Ix = 0

When does this

(A - I)x = 0

homogeneous system have a

solution other than x = 0 ?

Must have that A - I is not invertible, which means that det(A - I) = 0

eigenvalues of A

find values of such that det(A - I) = 0

given eigenvalue ,

associated eigenvectors are nonzero vectors in null(A - I)

Example of finding eigenvalues and eigenvectors

Example Find eigenvalues and

1 0 -1

corresponding eigenvectors of A. A = 2 -1 5

00 2

0

=

det(A

-

I)

=

1

- 2 0

0 -1 -

0

2--51

characteristic polynomial

-3 + 22 + - 2

=

(2

-

)

1

- 2

0 -1 -

= (2 - )(1 - )(-1 - )

= 2, 1, or - 1

Example of finding eigenvalues and eigenvectors

Example Find eigenvalues and corresponding eigenvectors of A.

= 2, 1, or - 1

1 0 -1

A = 2 -1 5

00 2

= 2 Solve (A - 2I)x = 0.

-1 0 -1 0

10 1 0

[ A - 2I | 0 ] = 2 -3 5 0 -E-R-Os 0 1 -1 0

0 0 00

00 0 0

x1

-x3

-1

x = x2 = x3 = x3 1

x3

x3

1

for any x3 R

eigenvectors of A for = 2 are -1

c 1 for c = 0

1

Example of finding eigenvalues and eigenvectors

Example Find eigenvalues and corresponding eigenvectors of A.

= 2, 1, or - 1

1 0 -1

A = 2 -1 5

00 2

=2

Solve (A - 2I)x = 0.

-1

eigenvectors of A for = 2 are c 1 for c = 0

1

E2

=

eigenspace of for = 2

A

=

set of all eigenvectors of A for = 2

{0}

= null(A - 2I) -1

= span 1

1

Example of finding eigenvalues and eigenvectors

Example Find eigenvalues and corresponding eigenvectors of A.

= 2, 1, or - 1

1 0 -1

A = 2 -1 5

00 2

=2 =1 = -1

-1

E2 = span 1 1

Solve (A - I)x = 0. =

Solve (A + I)x = 0. =

1

E1 = span 1 0

0

E-1 = span 1 0

Eigenvalues, eigenvalues... where are you?

Example Find eigenvalues of A.

11

A = -1 1

0 = det(A - I) = 1--1

1

1 -

= (1 - )2 + 1

= 2 - 2 + 2

2? = =

4-8

=1?i

2

Eigenvalues are complex!

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