How many matrices can be spectrally balanced simultaneously?

2016/06/06 by Eldan, Ronen, Nazarov, Fedor, Peres, Yuval
#FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.1606.01680

Abstract

We prove that any ℓ positive definite d × d matrices, M₁,…,M_ℓ, of full rank, can be simultaneously spectrally balanced in the following sense: for any k < d such that ℓ ≤ \lfloor (d-1)/(k-1) \rfloor, there exists a matrix A satisfying (λ₁(A^T M_i A) )/( Tr( A^T M_i A ) ) < (1)/(k) for all i, where λ₁(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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