Related papers: A characterization of trace zero symmetric nonnega…
For a graph $G$, we associate a family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location of the nonzero off-diagonal entries of $M$ are governed by the adjacency structure of $G$. The ordered…
Given two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. In this paper, we consider the following problem: given two parametric real symmetric matrices and an…
In this paper, we answer the various forms of nonnegative inverse eigenvalue problems with prescribed diagonal entries for order three: real or complex general matrices, symmetric stochastic matrices, and real or complex doubly stochastic…
We show that the structured singular value of a real matrix with respect to five full complex uncertainty blocks equals its convex upper bound. This is done by formulating the equality conditions as a feasibility SDP and invoking a result…
Exact evaluation of $<{\rm Tr} S^p>$ is here performed for real symmetric matrices $S$ of arbitrary order $n$, up to some integer $p$, where the matrix entries are independent identically distributed random variables, with an arbitrary…
Let $\mathcal{S}$ be the set of all positive-definite, symmetrizable integer matrices with non-zero upper and lower diagonal and $\mathcal{T}$ to be the set of all positive-definite real symmetric matrices with nonzero upper diagonal such…
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.…
In this article we develop new methods for exhibiting convex semialgebraic sets that are not spectrahedral shadows. We characterize when the set of nonnegative polynomials with a given support is a spectrahedral shadow in terms of sums of…
In this work, the real nonnegative inverse eigenvalue problem is solved for a particular class of permutative matrix. The necessary and sufficient condition there is also shown to be sufficient for the symmetric nonnegative inverse…
We explore the asymptotic convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. These are natural matrix analogs of scalar nonnegative…
A symmetric doubly stochastic matrix A is said to be determined by its spectra if the only symmetric doubly stochastic matrices that are similar to A are of the form $P^TAP$ for some permutation matrix P. The problem of characterizing such…
The inverse eigenvalue problem studies the possible spectra among matrices whose off-diagonal entries have their zero-nonzero patterns described by the adjacency of a graph $G$. In this paper, we refer to the $i$-nullity pair of a matrix…
The independence number of a square matrix $A$, denoted by $\alpha(A)$, is the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set of $n\times n$ nonnegative symmetric matrices with zero trace. Denote by $J_n$ the…
This note is concerned with isometries on the spaces of self-adjoint traceless matrices. We compute the group of isometries with respect to any unitary similarity invariant norm. This completes and extends the result of Nagy on Schatten…
The Nystr\"om method is a popular choice for finding a low-rank approximation to a symmetric positive semi-definite matrix. The method can fail when applied to symmetric indefinite matrices, for which the error can be unboundedly large. In…
All graphs considered are simple and undirected. The Inverse Eigenvalue Problem of a Graph $G$ (IEP-G) aims to find all possible spectra for matrices whose $(i,j)-$entry, for $i\neq j$, is nonzero precisely when $i$ is adjacent to $j$. A…
We consider the problem of finding nonzero eigenvalues and the corresponding eigenvectors of a matrix $AA^{\top}$, where $A$ is a special incidence matrix; This matrix can equivalently be defined based on a match relation between some…
Over the past decades, transformations between different classes of eigenvalue problems have played a central role in the development of numerical methods for eigenvalue computations. One of the most well-known and successful examples of…
The inverse eigenvalue problem of a graph studies the real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of the graph. The strong spectral property (SSP) is an important tool for this problem. This note…
We present numerical methods for computing the Schatten $p$-norm of positive semi-definite matrices. Our motivation stems from uncertainty quantification and optimal experimental design for inverse problems, where the Schatten $p$-norm…