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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
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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 AMnn:
det(I  A)  (I  A)  n  cn 1n 1    c1  c0
• 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 of a linear tra nsformatio n
T : V  V if there is a nonzero vector x such that T (x)  x.
The vector x is called an eigenvecto r of T correspond ing to  ,
and the setof all eigenvecto rs of  (with the zero vector) is
called the eigenspace 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 '  Pw '  y '  Ay  APw
1
 w '  P APw
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• Quadratic Forms
a' and c ' are eigenvalues of the matrix:
matrix of the quadratic form
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