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Chapter 7

Eigenvalues and Eigenvectors7.1 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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• Characteristic polynomial of AMnn:

011

1)I()Idet( cccAA nn

n

• Characteristic equation of A:

0)Idet( A

If has nonzero solutions iff . 0)I( xA 0)Idet( A0)I( xAxAx

• Note:(homogeneous system)

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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:

. of eigenspace thecalled

is vector)zero (with the of rseigenvecto all setof theand

, toingcorrespond ofr eigenvectoan called is vector The

.)(such that vector nonzero a is thereif :

n nsformatiolinear tra a of eigenvaluean called is number A

T

TVVT

x

xxx

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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 , then two square matrices A and B are called similar. (2) The eigenvalue problem is related closely to the diagonalization problem.

APPB 1

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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:

wy P

wyywwy APAPP ''''

ww APP 1'

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• Quadratic Forms

and are eigenvalues of the matrix:'a 'c

matrix of the quadratic form

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