A note on matrices mapping a positive vector onto its element-wise inverse

TL;DR

This paper proves the existence and uniqueness of a positive vector related to a primitive matrix with positive diagonal entries, offering an alternative proof of a classical result on matrix diagonal equivalence to stochastic matrices.

Contribution

It provides a new proof of a known result regarding the diagonal equivalence of nonnegative matrices to stochastic matrices.

Findings

Existence and uniqueness of the positive vector for primitive matrices.

Alternative proof of Brualdi et al.'s 1966 result.

Insight into matrix mappings involving element-wise inverses.

Abstract

For any primitive matrix $M\in\mathbb{R}^{n\times n}$ with positive diagonal entries, we prove the existence and uniqueness of a positive vector $\mathbf{x}=(x_1,\dots,x_n)^t$ such that $M\mathbf{x}=(\frac{1}{x_1},\dots,\frac{1}{x_n})^t$. The contribution of this note is to provide an alternative proof of a result of Brualdi et al. (1966) on the diagonal equivalence of a nonnegative matrix to a stochastic matrix.