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Mathematics-Online course: Linear Algebra - Normal Forms - Diagonalisation

Powers of Matrices

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Let $ A$ be a matrix having an eigenvalue $ \lambda$ of largest absolute value with associated eigenvector $ u$. If vector $ x$ has a nontrivial component in the eigenspace of $ \lambda$, that is, if $ x=cu+v$ with $ c\ne0$ and $ u\nparallel v$, then we have

$\displaystyle A^n x = \lambda^n (c u + o(1)),\quad n\to\infty \,. $

(Authors: App/Burkhardt/Höllig)

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  automatically generated 4/21/2005