Transcript CHAPTER 7
Chapter 7
Eigenvalues and Eigenvectors
7.1 Eigenvalues and eigenvectors
• Eigenvalue problem: If A
is an nn matrix, do there
exist nonzero vectors x in
Rn such that Ax is a scalar
multiple of x
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• 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 AMnn:
det( I A ) ( I A ) c n 1
n
n 1
c1 c 0
• Characteristic equation of A:
det( I A ) 0
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• 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.
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• 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 .
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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.
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7.3 Symmetric Matrices and Orthogonal
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• Note: Theorem 7.7 is called the Real Spectral Theorem, and the
set of eigenvalues of A is called the spectrum of A.
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• Note: A matrix A is orthogonally diagonalizable if there exists
an orthogonal matrix P such that P-1AP = D is diagonal.
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7.4 Applications of Eigenvalues and
Eigenvectors
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• 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
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• Quadratic Forms
a ' and c ' are eigenvalues of the matrix:
matrix of the quadratic form
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