1
$\begingroup$

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!

  • 0
    If the matrix is diagonalizable, this question is trivial.2012-06-14

1 Answers 1

1

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.

  • 0
    Can'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