Related papers: Perturbing eigenvalues of nonnegative centrosymmet…

We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues…

In this paper we express the eigenvalues of anti-heptadiagonal persymmetric Hankel matrices as the zeros of explicit polynomials giving also a representation of its eigenvectors. We present also an expression depending on localizable…

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible…

We give an almost-complete description of orthogonal matrices $M$ of order $n$ that "rotate a non-negligible fraction of the Boolean hypercube $C_n=\{-1,1\}^n$ onto itself," in the sense that $$P_{x\in C_n}(Mx\in C_n) \ge n^{-C},\mbox{ for…

Let $\pi(A)$, $\xi(A)$ and $\nu(A)$, respectively, denote the number of positive, zero and negative eigenvalues of the matrix $A$. Then the triplet $(\pi(A), \xi(A), \nu(A))$ is called the \emph{inertia} of $A$ and is denoted by…

We study the differential and metric structures of the set of real square roots of a non-singular real matrix, under the assumption that the matrix and its square roots are semi-simple, or symmetric, or orthogonal.

In this work we present a framework for studying the eigenvalues of a family of matrices with a particular displacement structure. The family admits a specific decomposition as the product of an upper and a lower triangular matrices having…

We present and compare three constructive methods for realizing non-real spectra with three nonzero elements in the nonnegative inverse eigenvalue problem. We also provide some necessary conditions for realizability and numerical examples.…

Symmetric kernel matrices are a well-researched topic in the literature of kernel based approximation. In particular stability properties in terms of lower bounds on the smallest eigenvalue of such symmetric kernel matrices are thoroughly…

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.…

Let $c_q(n)$ be the Ramanujan sums and let $S(m,n;q)$ be the Kloosterman sums. We study the eigenvalues of $q \times q$ matrices whose $(m,n)$ entry is $c_q (m-n)$ or $S(m,n;q)$ where $q$ is a fixed positive integer. We also study the…

Large H-selfadjoint random matrices are considered. The matrix $H$ is assumed to have one negative eigenvalue, hence the matrix in question has precisely one eigenvalue of nonpositive type. It is showed that this eigenvalue converges in…

We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real…

Nonclassical symmetries and reductions of polynomial equations and systems of polynomial equations are considered. It is shown that specific polynomial equations having "hidden" symmetries can be reduced to classical symmetric systems of…

We study a nonlinear decomposition of a positive definite matrix into two components: the inverse of another positive definite matrix and a symmetric matrix constrained to lie in a prescribed linear subspace. Equivalently, the inverse…

Recently, Zhao and Yang introduced centrosymmetric tensors. In this paper, we further introduce skew centrosymmetric tensors and centrosymmetric Cauchy tensors, and discuss properties of these three classes of structured tensors. Some…

A matrix is homogeneous if all of its entries are equal. Let $P$ be a $2\times 2$ zero-one matrix that is not homogeneous. We prove that if an $n\times n$ zero-one matrix $A$ does not contain $P$ as a submatrix, then $A$ has an $cn\times…

We analyze the spectral properties of correlation matrices between distinct statistical systems. Such matrices are intrinsically non symmetric, and lend themselves to extend the spectral analyses usually performed on standard Pearson…

Statistical properties of non--symmetric real random matrices of size $M$, obtained as truncations of random orthogonal $N\times N$ matrices are investigated. We derive an exact formula for the density of eigenvalues which consists of two…

Symmetrizable matrices are those which are symmetric when multiplied by a diagonal matrix with positive entries. The Cauchy interlace theorem states that the eigenvalues of a real symmetric matrix interlace with those of any principal…