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Mathematik-Online lexicon: Annotation to

Characteristic Polynomial

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The eigenvalues of matrix $ A$ are the zeros of the characteristic polynomial

$\displaystyle p_A(\lambda) = \operatorname{det}(A - \lambda E) \,. $

$ \lambda$ is an eigenvalue of the $ n\times n$-matrix $ A$ if and only if

$\displaystyle Av=\lambda v\,,v\neq0\,, $

that is, if the homogeneous linear system of equations $ (A-\lambda E)v=0$ has a non-zero solution. This is the case if and only if $ (A-\lambda E)$ is singular, that is, if $ \operatorname{det}(A-\lambda E) =0$.

(Authors: Burkhardt/Höllig/Hörner)
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  automatisch erstellt am 16.  3. 2005