Related papers: Refined Inertia of Matrix Patterns
Computing eigenvalues of very large matrices is a critical task in many machine learning applications, including the evaluation of log-determinants, the trace of matrix functions, and other important metrics. As datasets continue to grow in…
Determinants of structured matrices play a fundamental role in both pure and applied mathematics, with wide-ranging applications in linear algebra, combinatorics, coding theory, and numerical analysis. In this work, the enumeration of…
In a very recent paper "\emph{On eigenvalues and equivalent transformation of trigonometric matrices}" (D. Zhang, Z. Lin, and Y. Liu, LAA 436, 71--78 (2012)), the authors motivated and discussed a trigonometric matrix that arises in the…
In this paper, we investigate the non-symmetric Strong Spectral Property (nSSP) from a combinatorial perspective. To zero-nonzero patterns of matrices we associate directed graphs and study when they require or allow the nSSP, providing a…
Matrix concentration inequalities give bounds for the spectral-norm deviation of a random matrix from its expected value. These results have a weak dimensional dependence that is sometimes, but not always, necessary. This paper identifies…
We construct a free fermion and matrix model representation of refined BPS generating functions of D2 and D0 branes bound to a single D6 brane, in a class of toric manifolds without compact four-cycles. In appropriate limit we obtain a…
We present a simple, accurate method for solving consistent, rank-deficient linear systems, with or without addi- tional rank-completing constraints. Such problems arise in a variety of applications, such as the computation of the…
The study of solving inverse singular value problems for nonnegative matrices has been around for decades. It is clear that an inverse singular problem is trivial if the desirable matrix is not restricted to a certain structure. Provided…
Discrete structures are currently second-class in differentiable programming. Since functions over discrete structures lack overt derivatives, differentiable programs do not differentiate through them and limit where they can be used. For…
This paper deals with strong structural controllability of linear structured systems in which the system matrices are given by zero/nonzero/arbitrary pattern matrices. Instead of assuming that the nonzero and arbitrary entries of the system…
A hermitian matrix can be parametrized by a set consisting of its determinant and the eigenvalues of its submatrices. We established a group of equations which connect these variables with the mixing parameters of diagonalization. These…
It is an intriguing question to see what kind of information on the structure of an oriented graph $D$ one can obtain if $D$ does not contain a fixed oriented graph $H$ as a subgraph. The related question in the unoriented case has been an…
The spectral properties of a variety of improved staggered operators are studied in quenched QCD. The systematic dependence of the infrared eigenvalue spectrum on i) improvement in the staggered operator, ii) improvement in the gauge field…
Hermitian linear matrix pencils are ubiquitous in control theory, operator systems, semidefinite optimization, and real algebraic geometry. This survey reviews the fundamental features of the matricial solution set of a linear matrix…
Random matrices tend to be well conditioned, and we employ this well known property to advance matrix computations. We prove that our algorithms employing Gaussian random matrices are efficient, but in our tests the algorithms have…
We characterize the eigenvalues and eigenvectors of a class of complex valued tridiagonal $n$ by $n$ matrices subject to arbitrary boundary conditions, i.e. with arbitrary elements on the first and last rows of the matrix. %By boundary…
We investigate an approach to matroid complexity that involves describing a matroid via a list of independent sets, bases, circuits, or some other family of subsets of the ground set. The computational complexity of algorithmic problems…
One of the distinguishing characteristics of modern deep learning systems is that they typically employ neural network architectures that utilize enormous numbers of parameters, often in the millions and sometimes even in the billions.…
In the Inverse Matroid problem, we are given a matroid, a fixed basis $B$, and an initial weight function, and the goal is to minimally modify the weights -- measured by some function -- so that $B$ becomes a maximum-weight basis. The…
Let A be an m \times n matrix in which the entries of each row are all distinct. Drisko showed that, if m \ge 2n-1, then A has a transversal: a set of n distinct entries with no two in the same row or column. We generalize this to matrices…