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Mathematics-Online course: Linear Algebra - Normal Forms - Jordan Normal Form

Powers of Matrices

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The matrix powers $ A^n$, $ n=0,1,\ldots$, converge to the zero matrix if and only if the absolute value of each eigenvalue $ \lambda$ is smaller than $ 1$.

The sequence $ (A^n)$ is bounded if $ \vert\lambda\vert\le 1$ and algebraic and geometric multiplicity are equal for eigenvalues of modulus $ 1$.

If there exists an eigenvalue of modulus greater than $ 1$, then the sequence diverges.

(Authors: Burkhardt/Höllig/Hörner)
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  automatically generated 4/21/2005