Related papers: Generic canonical forms for perplectic and symplec…

Let $U_n$ be an $n \times n$ Haar unitary matrix. In this paper, the asymptotic normality and independence of $\Tr U_n, \Tr U_n^2, ..., \Tr U_n^k$ are shown by using elementary methods. More generally, it is shown that the renormalized…

A universal adjacency matrix of a graph $G$ with adjacency matrix $A$ is any matrix of the form $U = \alpha A + \beta I + \gamma J + \delta D$ with $\alpha \neq 0$, where $I$ is the identity matrix, $J$ is the all-ones matrix and $D$ is the…

In this note, we consider matrices similar to $X$-form matrices, which are the matrices for which only the diagonal and the anti-diagonal elements can be different from zero. First, we give a characterization of these matrices using the…

We study highly dissipative H\'enon maps $$ F_{c,b}: (x,y) \mapsto (c-x^2-by, x) $$ with zero entropy. They form a region $\Pi$ in the parameter plane bounded on the left by the curve $W$ of infinitely renormalizable maps. We prove that…

Let $D$ denote a positive integer and let $Q_D$ denote the graph of the $D$-dimensional hypercube. Let $X$ denote the vertex set of $Q_D$ and let $A \in \MX$ denote the adjacency matrix of $Q_D$. A matrix $B \in \MX$ is called $A$-{\em…

Let S=(s_1,s_2,..., s_m) and T = (t_1,t_2,..., t_n) be vectors of non-negative integers with sum_{i=1}^{m} s_i = sum_{j=1}^n t_j. Let B(S,T) be the number of m*n matrices over {0,1} with j-th row sum equal to s_j for 1 <= j <= m and k-th…

The Deligne-Simpson problem is formulated like this: give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset SL(n,{\bf C})$ or $c_j\subset sl(n,{\bf C})$ so that there exist irreducible $(p+1)$-tuples of…

Generalized diagonal matrices are matrices that have two ladders of entries that are zero in the upper right and bottom left corners. The minors of generic generalized diagonal matrices have square-free initial ideals. We give a description…

Recently much effort has been made towards the introduction of non-Hermitian random matrix models respecting $PT$-symmetry. Here we show that there is a one-to-one correspondence between complex $PT$-symmetric matrices and split-complex and…

Given a quantum permutation group $G\subset S_N^+$, with orbits having the same size $K$, we construct a universal matrix model $\pi:C(G)\to M_K(C(X))$, having the property that the images of the standard coordinates $u_{ij}\in C(G)$ are…

The standard format of matrices belonging to Lie superalgebras consists of partitioning the matrices into even and odd blocks. In this paper, we study other possible matrix formats and in particular the so-called diagonal format which…

Let H be a positive semidefinite matrix partitioned into Hermitian blocks. Then, up to a direct sum operation, H is the average of matrices isometrically congruent to its partial trace. A few corollaries are given, related to important…

Hermitian and unitary matrices are two representatives of the class of normal matrices whose full eigenvalue decomposition can be stably computed in quadratic computing com plexity. Recently, fast and reliable eigensolvers dealing with low…

A matrix $A\in \mathbb{R}^{m \times n}$ is strictly sign regular/SSR (or sign regular/SR) if for each $1 \leq k \leq \min\{m,n\}$, all (non-zero) $k\times k$ minors of $A$ have the same sign. This class of matrices contains the totally…

We consider the alternating sign matrices of the odd order that have some kind of central symmetry. Namely, we deal with matrices invariant under the half-turn, quarter-turn and flips in both diagonals. In all these cases, there are two…

The Jordan normal form for a matrix over an arbitrary field and the canonical form for a pair of matrices under contragredient equivalence are derived using Ptak's duality method.

We exhibit an explicit, deterministic algorithm for finding a canonical form for a positive definite matrix under unimodular integral transformations. We use characteristic sets of short vectors and partition-backtracking graph software.…

The Hermitian matrix model with general linear symmetry is argued to decouple into a finite unitary matrix model that contains metastable multidimensional lattice configurations and a fermion determinant. The simplest metastable state is a…

We consider the problem of designing a variety of "system guided" basis sets for quantum mechanical anharmonic oscillators. Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and…

Dual complex matrices have found applications in brain science. There are two different definitions of the dual complex number multiplication. One is noncommutative. Another is commutative. In this paper, we use the commutative definition.…