How many matrices can be spectrally balanced simultaneously?

TL;DR

This paper proves conditions under which multiple positive definite matrices can be simultaneously spectrally balanced, addressing a question related to the transience of self-interacting random walks and providing bounds in some cases.

Contribution

It establishes new conditions for simultaneous spectral balancing of matrices, solving a previously posed question and advancing understanding of self-interacting random walk transience.

Findings

Existence of a matrix A balancing multiple matrices spectrally under certain conditions.

Complete answer to a question posed by Peres, Popov, and Sousi.

Quantitative bounds on the transience of self-interacting random walks in some cases.

Abstract

We prove that any $\ell$ positive definite $d \times d$ matrices, $M_1,\ldots,M_\ell$, of full rank, can be simultaneously spectrally balanced in the following sense: for any $k < d$ such that $\ell \leq \lfloor \frac{d-1}{k-1} \rfloor$, there exists a matrix $A$ satisfying $\frac{\lambda_1(A^T M_i A) }{ \mathrm{Tr}( A^T M_i A ) } < \frac{1}{k}$ for all $i$, where $\lambda_1(M)$ denotes the largest eigenvalue of a matrix $M$. This answers a question posed by Peres, Popov and Sousi and completes the picture described in that paper regarding sufficient conditions for transience of self-interacting random walks. Furthermore, in some cases we give quantitative bounds on the transience of such walks.