Does $\lim_{n\to\infty} A^n = 0$, when the maximum of the norms of the eigenvalues of $A$ is less than $1$? Background: I am not familiar with matrix theory yet. Thanks in advance!
Does $\lim_{n\to\infty} A^n = 0$, when the maximum of the norm of eigenvalues of $A$ is less than 1?
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matrices
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0If the matrix is diagonalizable, this question is trivial. – 2012-06-14
1 Answers
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If you know that the spectral radius of $A$ is the limit, when $n$ goes to infinity, of $||A ^n ||^{1 / n}$, then this is immediate.
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0Can't we just say that since the eigenvalues of $A^n$ are $\lambda^n$ so they all approach $0$. the only matrix with $n$ eigenvalues $0$ is the zero matrix and so we are done ? – 2012-08-13