The convergence of power matrices

TL;DR

This paper investigates the conditions under which the powers of certain complex matrices with column sums equal to one converge to a nonzero limit, revealing that for dimensions three and higher, positivity and reality of entries are essential.

Contribution

It establishes necessary and sufficient conditions for the convergence of power matrices in a specific class of complex matrices, extending understanding of matrix power behavior.

Findings

Convergence occurs only when all entries are positive and real for dimensions d ≥ 3.

Provides explicit conditions for convergence of power matrices.

Identifies the role of positivity and reality in matrix power limits.

Abstract

For the class of $d\times d$ matrices $B=[b_{i,j}]$ with complex nonzero entries satisfying $\sum_{i=1}^{d}|b_{i,j}|=1$, we provide the conditions for the convergence of power matrices $B^n$ to a nonzero limit matrix. In particular, for $d\geq3$ we prove that the sequence of matrices $B^n$ converges to a nonzero limit matrix if and only if all the entries are positive and real.