3A.9 Matrix Eigenvalue Problem

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3A.9.1 Characteristic Polynomial

Consider a square matrix A: The polynomial

DðsÞ ¼ det½sI 2 A􀀉

is called the characteristic polynomial of A:

3A.9.2 Characteristic Equation

The polynomial equation

DðsÞ ¼ det½sI 2 A􀀉 ¼ 0

is called the characteristic equation of the square matrix A:

3A.9.3 Eigenvalues

The roots of the characteristic equation of a square matrix A are the eigenvalues of A: For an n £ n matrix,

there will be n eigenvalues.

3A.9.4 Eigenvectors

The eigenvalue problem of a square matrix A is given by

Av ¼ lv

Modal Analysis 3-55

© 2005 by Taylor & Francis Group, LLC

where the objective is to solve for l and the corresponding nontrivial (i.e., nonzero) solutions for v:

The problem can be expressed as

ðlI 2 AÞv ¼ 0

Note: If v is a solution of this equation, then any multiple of it, av; is also a solution. Hence, an

eigenvector is arbitrary up to a multiplication factor.

For a nontrivial (i.e., nonzero) solution to be possible for v; one must have

det½lI 2 A􀀉 ¼ 0

Since this is the characteristic equation of A; as defined above, it is clear that the roots of l are the

eigenvalues of A: The corresponding solutions for v are the eigenvectors of A: For an n £ n matrix, there

will be n eigenvalues and n corresponding eigenvectors.

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