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Mathematik-Online lexicon: Annotation to

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}$.
Let $ A v = \lambda v$ and $ k\ge0$, then $ A^k v = \lambda^k v$, and, more generally,

$\displaystyle \left(\sum_k c_k A^k\right) v = \left(\sum_k c_k \lambda^k\right) v \,. $

Further we have

$\displaystyle A v = \lambda v \Leftrightarrow v = \lambda A^{-1} v\,, $

if $ A$ is invertible.

If $ \lambda$ is not a zero of the polynomial in the denominator $ q$, then we have

$\displaystyle q(A) v = q(\lambda) v,\quad q(A)^{-1} v = q(\lambda)^{-1} v\,, $

consequently,

$\displaystyle q(A)^{-1} p(A) v = q(\lambda)^{-1} p(\lambda) v = p(A) q(A)^{-1} v $

as asserted.
(Authors: Burkhardt/Höllig/Hörner )
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  automatisch erstellt am 8.  7. 2004