Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag

Mathematics-Online lexicon:

Rational Functions of Matrices


A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

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 )

see also:


[Annotations]

  automatically generated 7/ 8/2004