Extensions of Yamamoto-Nayak's Theorem

TL;DR

This paper extends Nayak's theorem by proving the existence of limits involving matrix powers and products for complex matrices, and generalizes the results to real semisimple Lie groups.

Contribution

It generalizes Nayak's theorem to include products with nonsingular matrices and extends the results to real semisimple Lie groups.

Findings

Limit of |A^m|^{1/m} exists for complex matrices.

Limit of |BA^mC|^{1/m} exists and is independent of B.

Results are generalized to real semisimple Lie groups.

Abstract

A result of Nayak asserts that $\underset{m\to \infty}\lim |A^m|^{1/m}$ exists for each $n\times n$ complex matrix $A$, where $|A| = (A^*A)^{1/2}$, and the limit is given in terms of the spectral decomposition. We extend the result of Nayak, namely, we prove that the limit of $\underset{m\to \infty}\lim |BA^mC|^{1/m}$ exists for any $n\times n$ complex matrices $A$, $B$, and $C$ where $B$ and $C$ are nonsingular; the limit is obtained and is independent of $B$. We then provide generalization in the context of real semisimple Lie groups.