Related papers: Determinants of Interval Matrices
This paper presents a method for expressing the determinant of an N {\times} N complex block matrix in terms of its constituent blocks. The result allows one to reduce the determinant of a matrix with N^2 blocks to the product of the…
We present an algorithm for computing a spectral decomposition of an interval matrix as an enclosure of spectral decompositions of particular realizations of interval matrices. The algorithm relies on tight outer estimations of eigenvalues…
The interval numbers is the set of compact intervals of $\mathbb{R}$ with addition and multiplication operation, which are very useful for solving calculations where there are intervals of error or uncertainty, however, it lacks an…
The problem of finding the distance from a given $n \times n$ matrix polynomial of degree $k$ to the set of matrix polynomials having the elementary divisor $(\lambda-\lambda_0)^j, \, j \geqslant r,$ for a fixed scalar $\lambda_0$ and $2…
Due to their importance in both data analysis and numerical algorithms, low rank approximations have recently been widely studied. They enable the handling of very large matrices. Tight error bounds for the computationally efficient…
The number of non-negative integer matrices with given row and column sums appears in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations of various kinds. Here we…
We propose a penalized likelihood framework for estimating multiple precision matrices from different classes. Most existing methods either incorporate no information on relationships between the precision matrices, or require this…
In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual…
We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime $p$. In particular, in the integer case, we improve a recent bound…
We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the…
For a given reciprocal matrix A, we give a union of matrix intervals in which any consistent matrix obtained from an efficient vector for A lies, and, conversely, any consistent matrix in this union comes from an efficient vector for A. The…
The Interior-Point Methods are a class for solving linear programming problems that rely upon the solution of linear systems. At each iteration, it becomes important to determine how to solve these linear systems when the constraint matrix…
Arithmetic constraints on integer intervals are supported in many constraint programming systems. We study here a number of approaches to implement constraint propagation for these constraints. To describe them we introduce integer interval…
The object of this paper is to introduce a new and fascinating method of solving large linear equations, based on Cramer's rule or Gaussian elimination but employing Sylvester's determinant identity in its computation process. In addition,…
This paper improves previously known bounds on the determinant of 0-1 matrices where each row has fixed support size. This uses a method based on Scheinerman's, with new analyses to improve upon his conjectures.
In this note, we present a systematic method to explicitly compute the determinants and inverses for some generalized Hilbert matrices associated with orthogonal systems with explicit representations. We expressed the determinant, the…
We present a new, practical algorithm for computing the determinant of a non-singular dense, uniform matrix over Z; the aim is to achieve better practical efficiency, which is always at least as good as currently known methods. The…
This paper introduces a framework to study discrete optimization problems which are parametric in the following sense: their constraint matrices correspond to matrices over the ring $\mathbb{Z}[x]$ of polynomials in one variable. We…
A problem that is frequently encountered in a variety of mathematical contexts, is to find the common invariant subspaces of a single, or set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea…
In the first part of these notes, we review some of the recent developments in the study of the spectral properties of Wigner matrices. In the second part, we present a new proof of a Wegner estimate for the eigenvalues of a large class of…