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Let \(A\) be a square matrix. In the following, a sequence of matrices \(C_{j}\) converges if all of its elements converge.

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Let \(A\) be a square matrix. In the following, a sequence of matrices \(C_{j}\) converges if all of its elements converge.
Prove that the following are equivalent:
(i) \(A^{k} \rightarrow 0\) as \(k \rightarrow \infty\) [each of the elements of \(A^{k}\) converge to zero].
(ii) All the eigenvalues \(\lambda_{j}\) of \(A\) have \(\left|\lambda_{j}\right|<1\).
(iii) The matrix geometric series \(\sum_{0}^{\infty} A^{k}\) converges to \((I-A)^{-1}\).
in Mathematics by Platinum (101k points) | 312 views

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