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Rational Functions of Matrices

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If eigenvalue $ \lambda$ of matrix $ A$ is not a pole of a given rational function

$\displaystyle r(t) = \frac{p(t)}{q(t)} = \frac{a_0 + a_1 t + \cdots}{b_0 + b_1 t + \cdots} \,, $

then $ r(\lambda)$ is a eigenvalue of

$\displaystyle r(A) = q(A)^{-1} p(A) = p(A) q(A)^{-1} \,. $

In particular, $ \lambda^k$ is an eigenvalue of matrix power $ A^{k}$ and, provided that $ A$ is invertible, $ 1/\lambda$ is an eigenvalue of $ A^{-1}$.
(Authors: Burkhardt/Höllig/Hörner )

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[Annotations]
  automatically generated 7/ 8/2004