An n × n sign pattern matrix A is an inertially arbitrary pattern if for every nonnegative triple (n1, n2, n3) with n1 + n2 + n3 = n, there is a real matrix in the sign pattern class of A having inertia (n1, n2, n3). An n× n sign pattern matrix A is a spectrally arbitrary pattern if for any given real monic polynomial r(x) of degree n, there is a real matrix in the sign pattern class of A with ...

Integer moments of the spectral determinant |det(zI−W )|2 of complex random matrices W are obtained in terms of the characteristic polynomial of the Hermitian matrix WW ∗ for the class of matrices W = AU where A is a given matrix and U is random unitary. This work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results in this context...

We have obtained the exact asymptotics of the determinant det1≤r,s≤L h ` r+s−2 r−1 ́

This paper deenes and develops cycle indices for the nite classical groups. These tools are then applied to study properties of a random matrix chosen uniformly from one of these groups. Properties studied by this technique will include semisimplicity, regularity, regular semisimplicity, the characteristic polynomial, number of Jordan blocks, and average order of a matrix.

We investigate the second-order correlation function of the characteristic polynomial of a sample covariance matrix. Starting from an explicit formula for a generating function, we re-obtain several well-known kernels from random matrix theory.

Let Mn(q) be the set of all n× n matrices with entries in the finite field Fq. With asymptotic probability one, the characteristic polynomial of a random A ∈ Mn(q) does not have all its roots in Fq. Let Xn(A) be the degree of the splitting field of the characteristic polynomial of A, and let μn be the average degree: μn = 1 |Mn(q)| ∑

Given a nonnegative integer m and a finite collection A of linear forms on Qd, the arrangement of affine hyperplanes in Qd defined by the equations α(x) = k for α ∈ A and integers k ∈ [−m,m] is denoted by Am. It is proved that the coefficients of the characteristic polynomial of Am are quasi-polynomials inm and that they satisfy a simple combinatorial reciprocity law.

In this paper, we define the symmetric hyperdeterminant, eigenvalues and E-eigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvalues are roots of another one-dimensional polynomial. These two one-dimensional polynomials are associated with the symmetric hyperdeterminant. We call them t...

For the relevant literature and most of the definitions and conventions the reader is referred to [1]. The next section contains an explicit description of the (reduced) characteristic polynomial of L4(s, t), and it is shown that this polynomial admits a decomposition into two closely related factors of degree 2 each. In Section 3 it is first shown that at least one of these factors must be red...

Let M (n, n) be the set of all n × n matrices over a commutative ring with identity. Then the Cayley Hamilton Theorem states: Theorem. Let A ∈ M (n, n) with characteristic polynomial det(tI − A) = c 0 t n + c 1 t n−1 + c 2 t n−2 + · · · + c n. Then c 0 A n + c 1 A n−1 + c 2 A n−2 + · · · + c n I = 0. In this note we give a variation on a standard proof (see [1], for example). The idea is to use...