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Matrix pencils, or pairs of matrices, may be used in a variety of applications. In particular, a pair of matrices (E,A) may be interpreted as the differential equation E x' + A x = 0. Such an equation is invariant by changes of variables,…
We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the…
Low rank regularization, in essence, involves introducing a low rank or approximately low rank assumption for matrix we aim to learn, which has achieved great success in many fields including machine learning, data mining and computer…
A regular matrix pencil sE-A and its rank one perturbations are considered. We determine the sets in the extended complex plane which are the eigenvalues of the perturbed pencil. We show that the largest Jordan chains at each eigenvalue of…
We propose a rational QZ method for the solution of the dense, unsymmetric generalized eigenvalue problem. This generalization of the classical QZ method operates implicitly on a Hessenberg, Hessenberg pencil instead of on a Hessenberg,…
Although algebraic matroids were discovered in the 1930s, interest in them was largely dormant until their recent use in applications of algebraic geometry. Because nonlinear algebra is computationally challenging, it is easier to work with…
A degeneration of curves gives rise to an interesting relation between linear systems on curves and on graphs. In this paper, we consider the case of linear pencils and as an application, we obtain some results on pencils on real curves.
The aim of this paper is to present new upper bounds for the distance between a properly normalized permanent of a rectangular complex matrix and the product of the arithmetic means of the entries of its columns. It turns out that the…
This paper proposes a rational filtering domain decomposition technique for the solution of large and sparse symmetric generalized eigenvalue problems. The proposed technique is purely algebraic and decomposes the eigenvalue problem…
A generalized eigenvalue algorithm for tridiagonal matrix pencils is presented. The algorithm appears as the time evolution equation of a nonautonomous discrete integrable system associated with a polynomial sequence which has some…
This paper starts by deriving a factorization of the Loewner matrix pencil that appears in the data-driven modeling approach known as the Loewner framework and explores its consequences. The first is that the associated quadruple…
Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from…
A square matrix is called stochastic (or row-stochastic) if it is non-negative and has each row sum equal to unity. Here, we constitute an eigenvalue localization theorem for a stochastic matrix, by using its principal submatrices. As an…
Numerous interesting properties in nonlinear systems analysis can be written as polynomial optimization problems with nonconvex sum-of-squares problems. To solve those problems efficiently, we propose a sequential approach of local…
Locally recoverable codes (LRCs) were proposed for the recovery of data in distributed and cloud storage systems about nine years ago. A lot of progress on the study of LRCs has been made by now. However, there is a lack of general theory…
We study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i.e., maps taking lines to plane curves.
Many applications give rise to structured matrix polynomials. The problem of constructing structure-preserving strong linearizations of structured matrix polynomials is revisited in this work and in the forthcoming ones…
The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the…