Smoothed analysis of symmetric random matrices with continuous distributions

Brendan Farrell; Roman Vershynin

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arXiv · 1212.3531

Smoothed analysis of symmetric random matrices with continuous distributions

Brendan Farrell·Roman Vershynin

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Abstract

We study invertibility of matrices of the form $D+R$ where $D$ is an arbitrary symmetric deterministic matrix, and $R$ is a symmetric random matrix whose independent entries have continuous distributions with bounded densities. We show that $|(D+R)^{-1}| = O(n^2)$ with high probability. The bound is completely independent of $D$. No moment assumptions are placed on $R$; in particular the entries of $R$ can be arbitrarily heavy-tailed.

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Brendan Farrell, Roman Vershynin. 2012-12-14. Smoothed analysis of symmetric random matrices with continuous distributions. https://arxiv.org/abs/1212.3531

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