Chapter 7
Eigenvalues and Eigenvectors
7.1 Eigenvalues and eigenvectors
• Eigenvalue problem: If A
is an nn matrix, do there
exist nonzero vectors x in
Rn such that Ax is a scalar
multiple of x
7-1
7-2
• Note:
Ax   x 
( I  A) x  0
(homogeneous system)
If (  I  A ) x  0 has nonzero solutions iff det(  I  A )  0.
• Characteristic polynomial of AMnn:
det(  I  A )  (  I  A )    c n  1 
n
n 1
   c1   c 0
• Characteristic equation of A:
det(  I  A )  0
7-3
7-4
• Notes:
(1)
If an eigenvalue 1 occurs as a multiple root (k times) for
the characteristic polynomial, then 1 has multiplicity k.
(2) The multiplicity of an eigenvalue is greater than or equal
to the dimension of its eigenspace.
7-5
7-6
• Eigenvalues and eigenvectors of linear transformations:
A number
 is called an eigenvalue
T : V  V if there is a nonzero
vector x such that
The vector x is called an eigenvecto
and the setof all eigenvecto
called the eigenspace
of a linear tra nsformatio
n
T ( x )   x.
r of T correspond ing to  ,
rs of  (with the
zero vector) is
of  .
7-7
7.2 Diagonalization
• Diagonalization problem: For a square matrix A, does there
exist an invertible matrix P such that P-1AP is diagonal?
• Notes:
(1) If there exists an invertible matrix P such that B  P  1 AP ,
then two square matrices A and B are called similar.
(2) The eigenvalue problem is related closely to the
diagonalization problem.
7-8
7-9
7-10
7-11
7-12
7-13
7.3 Symmetric Matrices and Orthogonal
7-14
• Note: Theorem 7.7 is called the Real Spectral Theorem, and the
set of eigenvalues of A is called the spectrum of A.
7-15
7-16
7-17
• Note: A matrix A is orthogonally diagonalizable if there exists
an orthogonal matrix P such that P-1AP = D is diagonal.
7-18
7-19
7.4 Applications of Eigenvalues and
Eigenvectors
7-20
• If A is not diagonal:
-- Find P that diagonalizes A:
y  Pw
 y '  Pw '
 w' P
1
 P w '  y '  A y  AP w
AP w
7-21
• Quadratic Forms
a ' and c ' are eigenvalues of the matrix:
matrix of the quadratic form
7-22
7-23
7-24