Invertibility of symmetric random matrices

Invertibility of symmetric random matrices

We study n × n symmetric random matrices H, possibly discrete, with iid abovediagonal entries. We show that H is singular with probability at most exp(−nc), and ‖H−1‖ = O(√n). Furthermore, the spectrum of H is delocalized on the optimal scale o(n−1/2). These results improve upon a polynomial singularity bound due to Costello, Tao and Vu, and they generalize, up to constant factors, results of T...

Invertibility of Random Matrices Lecture Notes

Invertibility of Random Matrices: Norm of the Inverse

Let A be an n × n matrix, whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate. We prove that the operator norm of A−1 does not exceed Cn3/2 with probability close to 1.

Invertibility of Random Matrices: Unitary and Orthogonal Perturbations

We show that a perturbation of any fixed square matrix D by a random unitary matrix is well invertible with high probability. A similar result holds for perturbations by random orthogonal matrices; the only notable exception is when D is close to orthogonal. As an application, these results completely eliminate a hard-to-check condition from the Single Ring Theorem by Guionnet, Krishnapur and Z...

The Littlewood-offord Problem and Invertibility of Random Matrices

We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n−1/2, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in...