Spectral symmetry in conference matrices

A conference matrix of order n is an $$n\times n$$ n × n matrix C with diagonal entries 0 and off-diagonal entries $$\pm 1$$ ± 1 satisfying $$CC^\top =(n-1)I$$ C C ⊤ = ( n - 1 ) I . If C is symmetric, then C has a symmetric spectrum $$\Sigma $$ Σ (that is, $$\Sigma =-\Sigma $$ Σ = - Σ ) and eigenvalues $$\pm \sqrt{n-1}$$ ± n - 1 . We show that many principal submatrices of C also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.

A note on adaptivity in factorized approximate inverse preconditioning

The problem of solving large-scale systems of linear algebraic equations arises in a wide range of applications. In many cases the preconditioned iterative method is a method of choice. This paper deals with the approximate inverse preconditioning AINV/SAINV based on the incomplete generalized Gram–Schmidt process. This type of the approximate inverse preconditioning has been repeatedly used for matrix diagonalization in computation of electronic structures but approximating inverses is of an interest in parallel computations in general. Our approach uses adaptive dropping of the matrix entries with the control based on the computed intermediate quantities. Strategy has been introduced as a way to solve di cult application problems and it is motivated by recent theoretical results on the loss of orthogonality in the generalized Gram– Schmidt process. Nevertheless, there are more aspects of the approach that need to be better understood. The diagonal pivoting based on a rough estimation of condition numbers of leading principal submatrices can sometimes provide inefficient preconditioners. This short study proposes another type of pivoting, namely the pivoting that exploits incremental condition estimation based on monitoring both direct and inverse factors of the approximate factorization. Such pivoting remains rather cheap and it can provide in many cases more reliable preconditioner. Numerical examples from real-world problems, small enough to enable a full analysis, are used to illustrate the potential gains of the new approach.

Distance Matrix of a Class of Completely Positive Graphs: Determinant and Inverse

A real symmetric matrix A is said to be completely positive if it can be written as BBt for some (not necessarily square) nonnegative matrix B. A simple graph G is called a completely positive graph if every matrix realization of G that is both nonnegative and positive semidefinite is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. We compute a matrix 𝒭 such that the inverse of the distance matrix of a class of completely positive graphs is expressed a linear combination of the Laplacian matrix, a rank one matrix of all ones and 𝒭. This expression is similar to the existing result for trees. We also bring out interesting spectral properties of some of the principal submatrices of 𝒭.

Universal Behavior of the Corners of Orbital Beta Processes

There is a unique unitarily-invariant ensemble of $N\times N$ Hermitian matrices with a fixed set of real eigenvalues $a_1> \dots > a_N$. The joint eigenvalue distribution of the $(N-1)$ top-left principal submatrices of a random matrix from this ensemble is called the orbital unitary process. There are analogous matrix ensembles of symmetric and quaternionic Hermitian matrices that lead to the orbital orthogonal and symplectic processes, respectively. By extrapolation, on the dimension of the base field, of the explicit density formulas, we define the orbital beta processes. We prove the universal behavior of the virtual eigenvalues of the smallest $m$ principal submatrices, when $m$ is independent of $N$ and the eigenvalues $a_1> \dots > a_N$ grow linearly in $N$ and in such a way that the rescaled empirical measures converge weakly. The limiting object is the Gaussian beta corners process. As a byproduct of our approach, we prove a theorem on the asymptotics of multivariate Bessel functions.

On infinite-dimensional integer Hankel matrices

We construct a parametric family of infinite-dimensional integer Hankel matrices, each of matrices of this family has the following property: its principal submatrices are nonsingular, and the set of prime divisors of the determinants of the principal submatrices coincides with the set of all primes.

Extreme Spectra Realization by Nonsymmetric Tridiagonal and Nonsymmetric Arrow Matrices

We consider the following inverse extreme eigenvalue problem: given the real numbers {λ1j,λjj}j=1n and the real vector x(n)=x1,x2,…,xn, to construct a nonsymmetric tridiagonal matrix and a nonsymmetric arrow matrix such that {λ1j,λjj}j=1n are the minimal and the maximal eigenvalues of each one of their leading principal submatrices, and x(n),λn(n) is an eigenpair of the matrix. We give sufficient conditions for the existence of such matrices. Moreover our results generate an algorithmic procedure to compute a unique solution matrix.