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For the unitary ensembles of $N\times N$ Hermitian matrices associated with a weight function $w$ there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For…
We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n-dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For…
We propose a supplement matrix method for computing eigenvalues of a dual Hermitian matrix, and discuss its application in multi-agent formation control. Suppose we have a ring, which can be the real field, the complex field, or the…
Two known computation methods and one new computation method for matrix determinant over an integral domain are discussed. For each of the methods we evaluate the computation times for different rings and show that the new method is the…
Matrix rank and inertia optimization problems are a class of discontinuous optimization problems in which the decision variables are matrices running over certain matrix sets, while the ranks and inertias of the variable matrices are taken…
In this paper we present an extension of the removal lemma to integer linear systems over abelian groups. We prove that, if the $k$--determinantal of an integer $(k\times m)$ matrix $A$ is coprime with the order $n$ of a group $G$ and the…
Starting from known results, due to Y. Tian in [Ti; 00], referring to the real matrix representations of the real quaternions, in this paper we will investigate the left and right real matrix representations for the complex quaternions and…
We show that Wigner semi-circle law holds for Hermitian matrices with dependent entries, provided the deviation of the cumulants from the normalised Gaussian case obeys a simple power law bound in the size of the matrix. To establish this…
Due to their rich algebraic structures and various applications, circulant matrices have been of interest and continuously studied. In this paper, the notions of Binomial-related matrices have been introduced. Such matrices are circulant…
We describe combinatorial properties of the defining row of a circulant Hadamard matrix by exploiting its orthogonality to subsequent rows, and show how to exclude several particular forms of these matrices.
We review some convexity inequalities for Hermitian matrices an add one more to the list.
In this paper the relationship between the problem of constructing the ground state energy for the quantum quartic oscillator and the problem of computing mean eigenvalue of large positively definite random hermitean matrices is…
We define geometric matrix midranges for positive definite Hermitian matrices and study the midrange problem from a number of perspectives. Special attention is given to the midrange of two positive definite matrices before considering the…
We discuss the integrable boundary conditions for the one-dimensional (1D) Hubbard Model in the framework of the Quantum Inverse Scattering Method (QISM). We use the fermionic R-matrix proposed by Olmedilla et al. to treat the twisted…
In this paper we obtain $L^1$-weighted norms of classical orthogonal polynomials (Hermite, Laguerre and Jacobi polynomials) in terms of the zeros of these orthogonal polynomials; these expressions are usually known as quadrature rules. In…
While real Hamiltonian mechanics and Hermitian quantum mechanics can both be cast in the framework of complex canonical equations, their complex generalisations have hitherto been remained tangential. In this paper quaternionic and…
The paper develops elementary linear algebra methods to compute the determinants of the tensor symmetrizations of quadratic and hermitian forms over fields of good characteristic. Explicit results are given for the partitions $(n)$,…
In this article we present several necessary and sufficient conditions for the existence of Hermitian positive definite solutions of nonlinear matrix equations of the form $X^s + A^*X^{-t}A + B^*X^{-p}B = Q$, where $ s, t, p \geq 1$, $ A,…
The object of this paper is to investigate the certain results involving Bateman's matrix polynomials for integral index. We obtain some properties, integral representation and recurrence relations for hypergeometric matrix function. We…
We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of…