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arXiv · 1606.01680

How many matrices can be spectrally balanced simultaneously?

Ronen Eldan·Fedor Nazarov·Yuval Peres
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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.

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Ronen Eldan, Fedor Nazarov, Yuval Peres. 2016-06-06. How many matrices can be spectrally balanced simultaneously?. https://arxiv.org/abs/1606.01680

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