Some oscillation criteria are given for the second order matrix differential system Y ′′ +Q(t)Y = 0, where Y and Q are n× n real continuous matrix functions with Q(t) symmetric, t ∈ [t0,∞). These results improve oscillation criteria recently discovered by Erbe, Kong and Ruan by using a generalized Riccati transformation V (t) = a(t){Y ′(t)Y −1(t) + f(t)I}, where I is the n × n identity matrix, ...

We consider a generalization of the full symmetric Toda hierarchy where the matrix L̃ of the Lax pair is given by L̃ = LS, with a full symmetric matrix L and a nondegenerate diagonal matrix S. The key feature of the hierarchy is that the inverse scattering data includes a class of noncompact groups of matrices, such as O(p, q). We give an explicit formula for the solution to the initial value pro...

The topic of spacing distributions in random matrix ensembles is almost as old as the introduction of random matrix theory into nuclear physics. Both events can be traced back to Wigner in the mid 1950’s [37, 38]. Thus Wigner introduced the model of a large real symmetric random matrix, in which the upper triangular elements are independently distributed with zero mean and constant variance, fo...

The first infinite families of symmetric designswere obtained fromfinite projective geometries, Hadamardmatrices, and difference sets. In this paper we describe two general methods of constructing symmetric designs that give rise to the parameters of all other known infinite families of symmetric designs. The method of global decomposition produces an incidence matrix of a symmetric design as a...

Let G be a graph with adjacency matrix A and let F be a field. An F-matrix Q is a support matrix of G if A = [Q ̸=O], the zero-nonzero pattern of Q. If G has an invertible skew-symmetric support F-matrix S, the S-dual G of G is defined as the graph with adjacency matrix [S−1 ̸= O]. An analogous adjacency matrix dual, G has been examined in the literature for those bipartite graphs G with unique p...

Motivated by the symmetric version of matrix multiplication we study the plethysm $S^k(\mathfrak{sl}_n)$ of the adjoint representation $\mathfrak{sl}_n$ of the Lie group $SL_n$. In particular, we describe the decomposition of this representation into irreducible components for $k=3$, and find highest weight vectors for all irreducible components. Relations to fast matrix multiplication, in part...

The spread of a matrix is defined as the maximum distances between any two eigenvalues that matrix. In this paper we investigate maximization function on compact convex subset set real symmetric matrices. We provide some general results and further study maximizing problem Sn[a,b] (the matrices with entries restricted to interval [a,b]). particular develop by X. Zhan (see [13]), S. M. Fallat J....

We show that there exist real numbers λ1, λ2, . . . , λn that occur as the eigenvalues of an entry-wise nonnegative n-by-n matrix but do not occur as the eigenvalues of a symmetric nonnegative n-by-n matrix. This solves a problem posed by Boyle and Handelman, Hershkowitz, and others. In the process, recent work by Boyle and Handelman that solves the nonnegative inverse eigenvalue problem by app...

Over the past few years, symmetric positive definite matrices (SPD) have been receiving considerable attention from computer vision community. Though various distance measures have been proposed in the past for comparing SPD matrices, the two most widely-used measures are affine-invariant distance and log-Euclidean distance. This is because these two measures are true geodesic distances induced...

may be reduced to a single linear integral equation whose kernel is defined on a^x^a+n(b — a), a^s^a+n(b — a). Greggi§ considered a system of the form (1) and by use of the transformation introduced by Fredholm showed the form of the resolvent matrix for the system; for the symmetric system where Ku(x; s) =Kn(s; x) (i,j = \,2, ■ ■ ■ , n) he also stated theorems analogous to those proved by Schm...