Transcript Eigenvectors and Linear Transformations

Eigenvectors and Linear Transformations
• Recall the definition of similar matrices:
Let A and C be nn 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
x  Ax and w  Dw
are essentially the same when viewed from the proper
perspective. Of course, this is a huge breakthrough since
the mapping w  Dw 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.
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:VW 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
T (x)B'  MxB , and
M
T (b1 )B' T (b2 )B'

T (bn )B' .
• Example. Let B be the standard basis for R2, and let B' be the
basis for R2 given by

c1  

2
2
2
2

 22 
, c 2   2 .

 2 
If T is rotation by 45º counterclockwise, what is M?
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
T(x)B  TB xB .
• Example. Let T: P3  P3 be defined by
T(a0  a 1 t  a 2 t  a 3 t )  a 1  2a 2 t  3a 3 t .
2
3
1
2
This is the _____________ operator. Let B = B' = {1, t, t2, t3}.
??
??
[T ]B  
??

??
??
??
??
??
??
??
??
??
??
??
.
??

??
Similarity of two matrix representations: A  PCP1
x
Multiplication by A
Multiplication
by P–1
xB
Ax
Multiplication
by P
Multiplication by C
AxB
Here, the basis B of R n is formed from the columns of P.
A linear operator: geometric description
• Let T: R 2  R 2 be defined as follows: T(x) is the
reflection of x in the line y = x.
y
T(x)
x
x
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:
0 1
A  [T ]E  
.

1 0
• The eigenvalues of A are solutions of:
det (A  I)  0.
• We have
  1 
2
det 


 1  0.

 1  
• The eigenvalues of A are: +1 and –1.
A basis of eigenvectors of A
• Let
 2 2
 2 2
u
, v  
.
 2 2
 2 2 
• Since Au = u and Av = –v, it follows that B ={u, v} is a
2
basis for R consisting of eigenvectors of A.
• The matrix representation of T with respect to basis B:
1 0 
D  [T ]B  
.

0  1
Similarity of two matrix representations
• The change-of-coordinates matrix from B to the standard
basis is P where
 2 2  2 2
P
.
2 2
 2 2
• Note that P-1= PT and that the columns of P are u and v.
• Next,
 2 2  2 2 1 0   2 2
PDP  



2 2  0  1  2 2
 2 2
-1
• That is,
P[T ]B P1  TE .
2 2  0 1 
.


2 2 1 0
A particular choice of input vector w
• Let w be the vector with E coordinates given by
0
w  [w]E   .
 2
y
x
y
w
v
u
T(w)
x
Transforming the chosen vector w by T
• Let w be the vector chosen on the previous slide. We have
1
[w ] B   .
1
• The transformation w T(w) can be written as

w 

P 1
D
P
 2
0
1
1
          T (w)

2
1
 1
0



alias alibi
alias
• Note that
1
[T (w)]B   .
 1
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
a  b 
b a .

