section47 - Chapter 4 The Vector Space Rn MAT188H1F Lec03...

Chapter 4: The Vector Space R
n
MAT188H1F Lec03 Burbulla
Chapter 4 Lecture Notes
Fall 2007
Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla
Chapter 4: The Vector Space R
n
Chapter 4: The Vector Space R
n
4.7: Orthogonal Diagonalization
Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla

Chapter 4: The Vector Space R
n
4.7: Orthogonal Diagonalization
Digonalization Revisited
In terms of the concepts of Chapter 4, we can now say that an
n × n matrix A is diagonalizable if and only if R
n
has a basis
consisting of eigenvectors of A . Why? A is diagonalizable if and
only if there is an invertible matrix P and a diagonal matrix D
such that
D = P
- 1
AP .
The columns of P are eigenvectors of A , and P is invertible, so the
eigenvectors form a basis of R
n
. We can also say that eigenvectors
corresponding to distinct eigenvalues of A must be independent.
Why? Suppose λ
1
= λ
2
and that AX
1
= λ
1
X
1
, AX
2
= λ
2
X
2
,
X
i
= O . Suppose a
1
X
1
+ a
2
X
2
= O . Then
λ
1
( a
1
X
1
+ a
2
X
2
) = O
A ( a
1
X
1
+ a
2
X
2
) = O
⇔
λ
1
a
1
X
1
+ λ
1
a
2
X
2
= O
a
1
λ
1
X
1
+ a
2
λ
2
X
2
= O
Subtract the last two equations: ( λ
1
- λ
2
) a
2
X
2
= O ⇒ a
2
= 0 .
Then a
1
X
1
= O ⇒ a
1
= 0 . So X
1
, X
2
are linearly independent.
Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla
Chapter 4: The Vector Space R
n
4.7: Orthogonal Diagonalization
Diagonalization and Multiplicity
The multiplicity of an eigenvalue λ is the number of times the
factor x - λ divides the characteristic polynomial C
A
( x ) of A . For
example, if C
A
( x ) = x
2
( x - 1)
3
, then λ
1
= 0 has multiplicity 2 ,
and λ
2
= 1 has multiplicity 3 . In terms of multiplicity we can
completely characterize when a matrix A is diagonalizable: A is
diagonalizable if and only for each eigenvalue λ of A ,
dim E
λ
( A ) = m ,
where m is the multiplicity of λ. For example,
A =
1 2
0 1
is not diagonalizable, since
dim E
1
( A ) = dim null
0 1
0 0
= 1 = 2 .
Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla

Chapter 4: The Vector Space R
n
4.7: Orthogonal Diagonalization
Eigenvectors of Symmetric Matrices
Theorem: Let X
1
and X
2
be eigenvectors of the symmetric n × n
matrix A , corresponding to the distinct eigenvalues λ
1
and λ
2
,
respectively. Then X
1
and X
2
are orthogonal.