Full sign-invertibility and symplectic matrices

Full Sign-Invertibility and Symplectic Matrices

An n n sign pattern H is said to be sign-invertible if there exists a sign pattern H 1 (called the sign-inverse of H) such that, for all matrices A 2 Q(H), A 1 exists and A 1 2 Q(H 1). If, in addition, H 1 is sign-invertible (implying (H 1) 1 = H), H is said to be fully sign-invertible and (H;H 1) is called a sign-invertible pair. Given an n n sign pattern H, a Symplectic Pair in Q(H) is a pair...

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 Matrices of Field Elements

In this paper the theory of invertibility of matrices of field elements (see e.g. [5], [6]) is developed. The main purpose of this article is to prove that the left invertibility and the right invertibility are equivalent for a matrix of field elements. To prove this, we introduced a special transformation of matrix to some canonical forms. Other concepts as zero vector and base vectors of fiel...

U–turn alternating sign matrices, symplectic shifted tableaux and their weighted enumeration

Alternating sign matrices with a U–turn boundary (UASMs) are a recent generalization of ordinary alternating sign matrices. Here we show that variations of these matrices are in bijective correspondence with certain symplectic shifted tableaux that were recently introduced in the context of a symplectic version of Tokuyama’s deformation of Weyl’s denominator formula. This bijection yields a for...