Eigenvectors and Linear Transformations

• Recall the definition of similar matrices: Let A and C be nn matrices. We say that A is similar to C in case A = PCP-1 for some invertible matrix P.

• A square matrix A is diagonalizable if A is similar to a diagonal matrix D.

• An important idea of this section is to see that the mappings are essentially the same

when viewed from the proper perspective. Of course, this is a huge breakthrough since the mapping is quite simple and easy to understand. In some cases, we may have to settle for a matrix C which is simple, but not diagonal.

wwxx DA and

ww D

Similarity Invariants for Similar Matrices A and C

Property Description

Determinant A and C have the same determinant

Invertibility A is invertible <=> C is invertible

Rank A and C have the same rank

Nullity A and C have the same nullity

Trace A and C have the same trace

Characteristic Polynomial A and C have the same char. polynomial

Eigenvalues A and C have the same eigenvalues

Eigenspace dimension

If is an eigenvalue of A and C, then the eigenspace of A corresponding to and the eigenspace of C corresponding to have the same dimension.

The Matrix of a Linear Transformation wrt Given Bases

• Let V and W be n-dimensional and m-dimensional vector spaces, respectively. Let T:VW be a linear transformation. Let B = {b1, b2, ..., bn} and B' = {c1, c2, ..., cm} be ordered bases for V and W, respectively. Then M is the matrix representation of T relative to these bases where

• Example. Let B be the standard basis for R2, and let B' be the basis for R2 given by

If T is rotation by 45º counterclockwise, what is M?

and ,M)T( ' BB xx

.)T()T()T( M 'n'2'1 BBB bbb

. ,22

22

2

22

22

1

cc

Linear Transformations from V into V

• In the case which often happens when W is the same as V and B' is the same as B, the matrix M is called the matrix for T relative to B or simply, the B-matrix for T and this matrix is denoted by [T]B. Thus, we have

• Example. Let T: be defined by

This is the _____________ operator. Let B = B' = {1, t, t2, t3}.

.T)T( BBB xx

.ta3ta2a)tatataT(a 23

121

33

2210

.

????????

????????

????????

????????

T][

B

33 PP

Similarity of two matrix representations: 1PCPA

BB A

A

xx

xx

Here, the basis B of is formed from the columns of P.

Multiplication by A

Multiplication by C

Multiplication by P–1

Multiplication by P

nR

A linear operator: geometric description

• Let T: be defined as follows: T(x) is the reflection of x in the line y = x.

x

y

x

T(x)

22 RR

Standard matrix representation of T and its eigenvalues

• Since T(e1) = e2 and T(e2) = e1, the standard matrix representation A of T is given by:

• The eigenvalues of A are solutions of:

• We have

• The eigenvalues of A are: +1 and –1.

.01

10[T]A

E

0.I)A(det

.011

1det 2

A basis of eigenvectors of A

• Let

• Since Au = u and Av = –v, it follows that B ={u, v} is a basis for consisting of eigenvectors of A.

• The matrix representation of T with respect to basis B:

.22

22 ,

22

22

vu

.10

01T][

BD

2R

Similarity of two matrix representations

• The change-of-coordinates matrix from B to the standard basis is P where

• Note that P-1= PT and that the columns of P are u and v.

• Next,

• That is,

.2222

2222

P

.01

10

2222

2222

10

01

2222

22221

-PDP

.TT][ 1EB PP

A particular choice of input vector w

• Let w be the vector with E coordinates given by

x

y

u

T(w)

.2

0 ][

Eww

v

wxy

Transforming the chosen vector w by T

• Let w be the vector chosen on the previous slide. We have

• The transformation w T(w) can be written as

• Note that

.1

1 ][

Bw

)T(0

2P

1

1D

1

1P

2

0

1

ww

aliasalibialias

.1

1)][T(

Bw

What can we do if a given matrix A is not diagonalizable?

• Instead of looking for a diagonal matrix which is similar to A, we can look for some other simple type of matrix which is similar to A.

• For example, we can consider a type of upper triangular matrix known as a Jordan form (see other textbooks for more information about Jordan forms).

• If Section 5.5 were being covered, we would look for a matrix of the form

.

ab

ba