Related papers: Regularizing algorithm for mixed matrix pencils
Paul Van Dooren [Linear Algebra Appl. 27 (1979) 103-140] constructed an algorithm for the computation of all irregular summands in Kronecker's canonical form of a matrix pencil. The algorithm is numerically stable since it uses only unitary…
We give an algorithm that uses only unitary transformations and for each square complex matrix constructs a *congruent matrix that is a direct sum of a nonsingular matrix and singular Jordan blocks.
We introduce a new family of strong linearizations of matrix polynomials---which we call "block Kronecker pencils"---and perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any…
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,…
This paper describes an algorithm which computes the characteristic polynomial of a matrix over a field within the same asymptotic complexity, up to constant factors, as the multiplication of two square matrices. Previously, this was only…
We investigate determinants of random unitary pencils (with scalar or matrix coefficients), which generalize the characteristic polynomial of a single unitary matrix. In particular we examine moments of such determinants, obtained by…
It is widely known that the lower bound for the algorithmic complexity of square matrix multiplication resorts to at least $n^2$ arithmetic operations. The justification builds upon the following reasoning: given that there are $2 n^2$…
In this paper we propose an approach to approximate a truncated singular value decomposition of a large structured matrix. By first decomposing the matrix into a sum of Kronecker products, our approach can be used to approximate a large…
The matrix pencil method is an eigenvalue based approach for the parameter identification of sparse exponential sums. We derive a reconstruction algorithm for multivariate exponential sums that is based on simultaneous diagonalization.…
V. I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] constructed a simple normal form for a family of complex n-by-n matrices that smoothly depend on parameters with respect to similarity transformations that smoothly depend on the same…
We propose a contour integral-based algorithm for computing a few singular values of a matrix or a few generalized singular values of a matrix pair. Mathematically, the generalized singular values of a matrix pair are the eigenvalues of an…
The multiparameter matrix pencil problem (MPP) is a generalization of the one-parameter MPP: given a set of $m\times n$ complex matrices $A_0,\ldots, A_r$, with $m\ge n+r-1$, it is required to find all complex scalars…
We exhibit an explicit, deterministic algorithm for finding a canonical form for a positive definite matrix under unimodular integral transformations. We use characteristic sets of short vectors and partition-backtracking graph software.…
The author was encouraged to write this review by numerous enquiries from researchers all over the world, who needed a ready-to-use algorithm for the inversion of confluent Vandermonde matrices which works in quadratic time for any values…
Let A, B, C, D be given finite sets of pairs of n-by-n complex matrices. We describe an algorithm to determine, with finitely many computations, whether there is a single unitary matrix U such that each pair of matrices in A is unitarily…
In 1989 we proposed to employ Vandermonde and Hankel multipliers to transform into each other the matrix structures of Toeplitz, Hankel, Vandermonde and Cauchy types as a means of extending any successful algorithm for the inversion of…
We introduce a new class of structured matrix polynomials, namely, the class of M_A-structured matrix polynomials, to provide a common framework for many classes of structured matrix polynomials that are important in applications: the…
We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of…
Compound matrices play an important role in many fields of mathematics and have recently found new applications in systems and control theory. However, the explicit formulas for these compounds are non-trivial and not always easy to use.…
The seminal work by Mackey et al. in 2006 (reference [21] of the article) introduced vector spaces of matrix pencils, with the property that almost all the pencils in the spaces are strong linearizations of a given square regular matrix…