Eigenvectors and Linear Transformations

1
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 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.

2
Similarity Invariants for Similar Matrices A and C
Property
Description
Determinant A and C
have the same determinant
Invertibility A is
invertible ltgt 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.
3
The Matrix of a Linear Transformation wrt Given
Bases

• Let V and W be n-dimensional and m-dimensional
  vector spaces, respectively. Let TV?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
• 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?

4
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 TB. Thus, we have
• Example. Let T be defined
  by This is the
  _____________ operator. Let B B' 1, t, t2,
  t3.

5
Similarity of two matrix representations
Multiplication by A
Multiplication by P1
Multiplication by P
Multiplication by C
Here, the basis B of is formed from the
columns of P.
6
A linear operator geometric description

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

y
T(x)
x
x
7
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.

8
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

9
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,

10
A particular choice of input vector w

• Let w be the vector with E coordinates given by

y
x?
y?
w
v
u
x
T(w)
11
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

12
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