Related papers: Rayleigh quotient and left eigenvalues of quaterni…
We extend previous work on the eigenvalue problem for Hermitian octonionic matrices by discussing the case where the eigenvalues are not real, giving a complete treatment of the 2x2 case, and summarizing some prelimenary results for the 3x3…
We give a minimal list of inequalities characterizing the possible eigenvalues of a set of Hermitian matrices with positive semidefinite sum of bounded rank. This answers a question of A. Barvinok.
The eigenvalue method, suggested by the developer of the extensively used Analytic Hierarchy Process methodology, exhibits right-left asymmetry: the priorities derived from the right eigenvector do not necessarily coincide with the…
We study a non-linear eigenvalue problem for vector-valued eigenfunctions and give a succinct uniqueness proof for minimizers of the associated Rayleigh quotient.
A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…
Horn's problem was the following: given two Hermitian matrices with known spectra, what might be the eigenvalue spectrum of the sum? This linear algebra problem is exactly of the sort to be approached with the methods of modern Hamiltonian…
In this paper, we investigate the properties of q-Hermite polynomials related to q-Bernstein polynomials. From these properties, we derive some interesting relations between q-Berstein polynomials and q-Hermite polynomials.
This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in (sturm2003cones,huang2007complex,ai2011new). The enhanced property can be used to…
We consider a constant-size subset of left and right eigenvectors of an $N\times N$ i.i.d. complex non-Hermitian matrix associated with the eigenvalues with pairwise distances at least $N^{-\frac12+\epsilon}$. We show that arbitrary…
We study analogues of classical inequalities for the eigenvalues of sums of pseudo-Hermitian matrices.
A recursive method is derived to calculate all eigenvalue correlation functions of a random hermitian matrix in the large size limit, and after smoothing of the short scale oscillations. The property that the two-point function is…
In noncommutative and nondivision algebra, left spectrum of matrices are less known and is not easy to handle. Split quaternion algebra is a noncommutative and nondivision algebra. In this paper, by the formulas of solving the equations…
We describe properties of a Hermitian square matrix M in M_n(C) equivalent to that of having minimal quotient norm in the following sense: ||M|| <= ||M+D|| for all real diagonal matrices D in M_n(C) and || || the operator norm. These…
Given a right eigenvector $x$ and a left eigenvector $y$ associated with the same eigenvalue of a matrix $A$, there is a Hermitian positive definite matrix $H$ for which $y=Hx$. The matrix $H$ defines an inner product and consequently also…
Properties of Hermitian forms are used to investigate several natural questions from CR Geometry. To each Hermitian symmetric polynomial we assign a Hermitian form. We study how the signature pairs of two Hermitian forms behave under the…
Dual quaternion matrices have various applications in robotic research and its spectral theory has been extensively studied in recent years. In this paper, we extend Jacobi method to compute all eigenpairs of dual quaternion Hermitian…
The eigenvalue problem for 3x3 octonionic Hermitian matrices contains some surprises, which we have reported elsewhere. In particular, the eigenvalues need not be real, there are 6 rather than 3 real eigenvalues, and the corresponding…
We compute quaisideterminants and determinants of quaternionic matrices
We use a quaternionic structure on the product of two symplectic manifolds for relating Liouvillian forms with linear symplectic maps obtained by the symplectic Cayley's transformation.
We study statistical properties of the eigenvectors of non-Hermitian random matrices, concentrating on Ginibre's complex Gaussian ensemble, in which the real and imaginary parts of each element of an N x N matrix, J, are independent random…