Invertibility of symmetric random matrices

Smoothed Analysis of Symmetric Random Matrices with Continuous Distributions Brendan Farrell and Roman Vershynin

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) 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 a...

Smoothed Analysis of Symmetric Random Matrices with Continuous Distributions

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) 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 a...

Invertibility and Largest Eigenvalue of Symmetric Matrix Signings

The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of identifying symmetric signings of matrices with natural spectral properties. Our results are twofold: 1. We show NP-completeness for the following three problems: verifying whether a given matrix has a symmetric signing th...

Invertibility of Random Matrices Lecture Notes

Some Results on Symmetric

In this work, we investigate several natural computational problems related to identifying symmetric signings of symmetric matrices with specific spectral properties. We show NP-completeness for verifying whether an arbitrary matrix has a symmetric signing that is positive semi-definite, is singular, or has bounded eigenvalues. We exhibit a stark contrast between invertibility and the above-men...