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linear algebra – Recover approximate monotonicity of induced norms Answer

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linear algebra – Recover approximate monotonicity of induced norms

Let $A$ some sq. matrix with actual entries.
Take any norm $|cdot|$ in line with a vector norm.

Gelfand’s formulation tells us that $rho(A) = lim_{n rightarrow infty} |A^n|^{1/n}$.

Moreover, from [1], for a sequence of $(n_i)_{i in mathbb{N}}$ such that $n_i$ is divisible by $n_{i-1}$, we too know that the sequence $|A^{n_i}|^{1/n_i}$ is monotone lowering and converges in direction of $rho(A)$. I’m thinking about what occurs when this divisibility property shouldn’t be verified.

  1. If the matrix has non-negative entries, it appears the common property holds: For integers $n$ and $m$ such that $m > n$, it’s the illustration that $|A^m|^{1/m} leq |A^n|^{1/n}$.

  2. If the matrix can have constructive and unfavorable entries, this extra common statement doesn’t appear to carry. I’m attempting to grasp why it fails, how worse can the inequality turn out to be, and whether it is workable to regain an inequality as much as some duty of $A$: $|A^m|^{1/m} leq f(A)cdot|A^n|^{1/n}$.

Any references to 1., or pointers for judgement 2. would breathe mighty appreciated.

[1] Yamamoto, Tetsuro. “On the extreme values of the roots of matrices.” Journal of the Mathematical Society of Japan 19.2 (1967): 173-178.

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