arXiv subjects

Froilán M. Dopico

Publications and source records attributed to Froilán M. Dopico.

At least 19 recordsLinked to original sources

arXiv2026

Distance to nearest skew-symmetric matrix polynomials of bounded rank

We propose an algorithm that approximates a given matrix polynomial of degree $d$ by another skew-symmetric matrix polynomial of a specified rank and degree at most $d$. The algorithm is built on recent advances in the theory of generic eigenstructures and factorizations for skew-symmetric matrix polynomials of bounded rank and degree. Taking into account that the rank of a skew-symmetric matrix polynomial is even, the algorithm works for any prescribed even rank greater than or equal to $2$ and produces a skew-symmetric matrix polynomial of that exact rank. We also adapt the algorithm for matrix pencils to achieve a better performance. Lastly, we present numerical experiments for testing our algorithms and for comparison to the previously known ones.

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arXiv2026

Generic real Jordan canonical forms

We obtain the generic real Jordan canonical forms for $n\times n$ matrices with real entries. More precisely, we prove that the set of $n\times n$ real matrices is the union of the closures of $\lfloor n/2\rfloor+1$ sets, which are called generic bundles, as they are particular "bundles". In general, a bundle is the set of $n\times n$ real matrices with the same real Jordan canonical form, up to the values of the eigenvalues, provided that the eigenvalues which are distinct in one matrix of the bundle remain distinct in any other matrix of the same bundle. The $k$th generic bundle, for $0\leq k\leq\lfloor n/2\rfloor$, contains the $n\times n$ real matrices having $k$ different pairs of non-real conjugate eigenvalues and $n-2k$ different real eigenvalues. We prove that each of the $\lfloor n/2\rfloor+1$ generic bundles is an open subset of the set of $n\times n$ real matrices. Some numerical experiments are carried out with large sets of random matrices of different sizes to confirm that all the generic bundles show up, and only these ones.

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arXiv2025

The generic canonical form for $^\star$congruence of matrices

First, we prove that the set of $n\times n$ complex matrices is the closure of a certain open subset whose elements have a very specific canonical form under congruence, which is uniquely determined up to the values of some parameters, but which has a slightly different expression depending on whether $n$ is even or odd. As a consequence, the canonical form under congruence of the elements of this subset can be considered the generic canonical form under congruence of complex $n\times n$ matrices. Second, we prove that the set of $n\times n$ complex matrices is the union of the closures of certain $\lfloor n/2\rfloor+1$ open subsets and that, for each of these subsets, its elements have a very specific canonical form under $^*$congruence, which is uniquely determined up to the values of some parameters. As a consequence, the $\lfloor n/2\rfloor+1$ canonical forms under $^*$congruence of the elements of each of these subsets can be considered the generic canonical forms under $^*$congruence of complex $n\times n$ matrices. So, there is only one generic canonical form under congruence whereas the number of generic canonical forms under $^*$congruence is $\lfloor n/2\rfloor+1$ instead, which reveals a strong dichotomy between the relations of congruence and $^*$congruence with respect to generic structures. In other words, we determine in this paper the generic matrix representations of $n\times n$ bilinear and sesquilinear forms in $\mathbb{C}^n \times \mathbb{C}^n$.

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arXiv2025

Real-congruence canonical forms of real matrices

We present two new canonical forms for real congruence of a real square matrix $A$. The first one is a direct sum of canonical matrices of four different types and is obtained from the canonical form under $^*$congruence of complex matrices provided by Horn and Sergeichuk in [Linear Algebra Appl. 416 (2006) 1010-1032]. The second one is a direct sum of canonical matrices of three different types, has a block tridiagonal structure and is obtained from the canonical form under $^*$congruence of complex matrices provided by Futorny, Horn and Sergeichuk in [J. Algebra 319 (2008) 2351-2371]. A detailed comparison between both canonical forms is also presented, as well as their relation with the real Kronecker canonical form under strict real equivalence of the matrix pair $(A^\top , A)$. Another canonical form for real congruence was presented by Lee and Weinberg in [Linear Algebra Appl. 249 (1996) 207-215], which consists of a direct sum of eight different types of matrices. In the last part of the paper, we explain the correspondence between the blocks in this canonical form and those in the two new forms introduced in this work.

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arXiv2025

On the dimension of orbits of matrix pencils under strict equivalence

We prove that, given two matrix pencils $L$ and $M$, if $M$ belongs to the closure of the orbit of $L$ under strict equivalence, then the dimension of the orbit of $M$ is smaller than or equal to the dimension of the orbit of $L$, and the equality is only attained when $M$ belongs to the orbit of $L$. Our proof uses only the majorization involving the eigenstructures of $L$ and $M$ which characterizes the inclusion relationship between orbit closures, together with the formula for the codimension of the orbit of a pencil in terms of its eigenstruture.

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arXiv2025

Polynomial and rational matrices with the invariant rational functions and the four sequences of minimal indices prescribed

The complete eigenstructure, or structural data, of a rational matrix $R(s)$ is comprised by its invariant rational functions, both finite and at infinity, which in turn determine its finite and infinite pole and zero structures, respectively, and by the minimal indices of its left and right null spaces. These quantities arise in many applications and have been thoroughly studied in numerous references. However, other two fundamental subspaces of $R(s)$ in contrast have received much less attention: the column and row spaces, which also have their associated minimal indices. This work solves the problems of finding necessary and sufficient conditions for the existence of rational matrices in two scenarios: (a) when the invariant rational functions and the minimal indices of the column and row spaces are prescribed, and (b) when the complete eigenstructure together with the minimal indices of the column and row spaces are prescribed. The particular, but extremely important, cases of these problems for polynomial matrices are solved first and are the main tool for solving the general problems. The results in this work complete and non-trivially extend the necessary and sufficient conditions recently obtained for the existence of polynomial and rational matrices when only the complete eigenstructure is prescribed.

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arXiv2024

On bundle closures of matrix pencils and matrix polynomials

Bundles of matrix polynomials are sets of matrix polynomials with the same size and grade and the same eigenstructure up to the specific values of the eigenvalues. It is known that the closure of the bundle of a pencil $L$ (namely, a matrix polynomial of grade $1$), denoted by $\mathcal{B}(L)$, is the union of $\mathcal{B}(L)$ itself with a finite number of other bundles. The first main contribution of this paper is to prove that the dimension of each of these bundles is strictly smaller than the dimension of $\mathcal{B}(L)$. The second main contribution is to prove that also the closure of the bundle of a matrix polynomial of grade larger than 1 is the union of the bundle itself with a finite number of other bundles of smaller dimension. To get these results we obtain a formula for the (co)dimension of the bundle of a matrix pencil in terms of the Weyr characteristics of the partial multiplicities of the eigenvalues and of the (left and right) minimal indices, and we provide a characterization for the inclusion relationship between the closures of two bundles of matrix polynomials of the same size and grade.

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arXiv2023

Even grade generic skew-symmetric matrix polynomials with bounded rank

We show that the set of $m \times m$ complex skew-symmetric matrix polynomials of even grade $d$, i.e., of degree at most $d$, and (normal) rank at most $2r$ is the closure of the single set of matrix polynomials with certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic $m \times m$ complex skew-symmetric matrix polynomials of even grade $d$ and rank at most $2r$. The analogous problem for the case of skew-symmetric matrix polynomials of odd grade is solved in [Linear Algebra Appl., 536:1-18, 2018].

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arXiv2022

Linearizations of matrix polynomials viewed as Rosenbrock's system matrices

A well known method to solve the Polynomial Eigenvalue Problem (PEP) is via linearization. That is, transforming the PEP into a generalized linear eigenvalue problem with the same spectral information and solving such linear problem with some of the eigenvalue algorithms available in the literature. Linearizations of matrix polynomials are usually defined using unimodular transformations. In this paper we establish a connection between the standard definition of linearization for matrix polynomials introduced by Gohberg, Lancaster and Rodman and the notion of polynomial system matrix introduced by Rosenbrock. This connection gives new techniques to show that a matrix pencil is a linearization of the corresponding matrix polynomial arising in a PEP.

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arXiv2022

Generic eigenstructures of Hermitian pencils

We obtain the generic complete eigenstructures of complex Hermitian $n\times n$ matrix pencils with rank at most $r$ (with $r\leq n$). To do this, we prove that the set of such pencils is the union of a finite number of bundle closures, where each bundle is the set of complex Hermitian $n\times n$ pencils with the same complete eigenstructure (up to the specific values of the finite eigenvalues). We also obtain the explicit number of such bundles and their codimension. The cases $r=n$, corresponding to general Hermitian pencils, and $r<n$ exhibit surprising differences, since for $r<n$ the generic complete eigenstructures can contain only real eigenvalues, while for $r=n$ they can contain real and non-real eigenvalues. Moreover, we will see that the sign characteristic of the real eigenvalues plays a relevant role for determining the generic eigenstructures of Hermitian pencils.

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arXiv2022

On bundles of matrix pencils under strict equivalence

Bundles of matrix pencils (under strict equivalence) are sets of pencils having the same Kronecker canonical form, up to the eigenvalues (namely, they are an infinite union of orbits under strict equivalence). The notion of bundle for matrix pencils was introduced in the 1990's, following the same notion for matrices under similarity, introduced by Arnold in 1971, and it has been extensively used since then. Despite the amount of literature devoted to describing the topology of bundles of matrix pencils, some relevant questions remain still open in this context. For example, the following two: (a) provide a characterization for the inclusion relation between the closures (in the standard topology) of bundles; and (b) are the bundles open in their closure? The main goal of this paper is providing an explicit answer to these two questions. In order to get this answer, we also review and/or formalize some notions and results already existing in the literature. We also prove that bundles of matrices under similarity, as well as bundles of matrix polynomials (defined as the set of $m\times n$ matrix polynomials of the same grade having the same spectral information, up to the eigenvalues) are open in their closure.

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arXiv2021

Quasi-Triangularization of Matrix Polynomials over Arbitrary Fields

In [19], Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial $P(λ)$ over an algebraically closed field is spectrally equivalent to a triangular matrix polynomial of the same degree. When $P(λ)$ is real and regular, they also show that there is a real quasi-triangular matrix polynomial of the same degree that is spectrally equivalent to $P(λ)$, in which the diagonal blocks are of size at most $2 \times 2$. This paper generalizes these results to regular matrix polynomials $P(λ)$ over arbitrary fields $\mathbb{F}$, showing that any such $P(λ)$ can be quasi-triangularized to a spectrally equivalent matrix polynomial over $\mathbb{F}$ of the same degree, in which the largest diagonal block size is bounded by the highest degree appearing among all of the $\mathbb{F}$-irreducible factors in the Smith form for $P(λ)$.

math.RA↗

arXiv2021

Strongly minimal self-conjugate linearizations for polynomial and rational matrices

We prove that we can always construct strongly minimal linearizations of an arbitrary rational matrix from its Laurent expansion around the point at infinity, which happens to be the case for polynomial matrices expressed in the monomial basis. If the rational matrix has a particular self-conjugate structure we show how to construct strongly minimal linearizations that preserve it. The structures that are considered are the Hermitian and skew-Hermitian rational matrices with respect to the real line, and the para-Hermitian and para-skew-Hermitian matrices with respect to the imaginary axis. We pay special attention to the construction of strongly minimal linearizations for the particular case of structured polynomial matrices. The proposed constructions lead to efficient numerical algorithms for constructing strongly minimal linearizations. The fact that they are valid for {\em any} rational matrix is an improvement on any other previous approach for constructing other classes of structure preserving linearizations, which are not valid for any structured rational or polynomial matrix. The use of the recent concept of strongly minimal linearization is the key for getting such generality.

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arXiv2021

Diagonal scalings for the eigenstructure of arbitrary pencils

In this paper we show how to construct diagonal scalings for arbitrary matrix pencils $λB-A$, in which both $A$ and $B$ are complex matrices (square or nonsquare). The goal of such diagonal scalings is to "balance" in some sense the row and column norms of the pencil. We see that the problem of scaling a matrix pencil is equivalent to the problem of scaling the row and column sums of a particular nonnegative matrix. However, it is known that there exist square and nonsquare nonnegative matrices that can not be scaled arbitrarily. To address this issue, we consider an approximate embedded problem, in which the corresponding nonnegative matrix is square and can always be scaled. The new scaling methods are then based on the Sinkhorn-Knopp algorithm for scaling a square nonnegative matrix with total support to be doubly stochastic or on a variant of it. In addition, using results of U. G. Rothblum and H. Schneider (1989), we give simple sufficient conditions on the zero pattern for the existence of diagonal scalings of square nonnegative matrices to have any prescribed common vector for the row and column sums. We illustrate numerically that the new scaling techniques for pencils improve the accuracy of the computation of their eigenvalues.

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arXiv2021

Structural backward stability in rational eigenvalue problems solved via block Kronecker linearizations

We study the backward stability of running a backward stable eigenstructure solver on a pencil $S(λ)$ that is a strong linearization of a rational matrix $R(λ)$ expressed in the form $R(λ)=D(λ)+ C(λI_\ell-A)^{-1}B$, where $D(λ)$ is a polynomial matrix and $C(λI_\ell-A)^{-1}B$ is a minimal state-space realization. We consider the family of block Kronecker linearizations of $R(λ)$, which are highly structured pencils. Backward stable eigenstructure solvers applied to $S(λ)$ will compute the exact eigenstructure of a perturbed pencil $\widehat S(λ):=S(λ)+Δ_S(λ)$ and the special structure of $S(λ)$ will be lost. In order to link this perturbed pencil with a nearby rational matrix, we construct a strictly equivalent pencil $\widetilde S(λ)$ to $\widehat S(λ)$ that restores the original structure, and hence is a block Kronecker linearization of a perturbed rational matrix $\widetilde R(λ) = \widetilde D(λ)+ \widetilde C(λI_\ell- \widetilde A)^{-1} \widetilde B$, where $\widetilde D(λ)$ is a polynomial matrix with the same degree as $D(λ)$. Moreover, we bound appropriate norms of $\widetilde D(λ)- D(λ)$, $\widetilde C - C$, $\widetilde A - A$ and $\widetilde B - B$ in terms of an appropriate norm of $Δ_S(λ)$. These bounds may be inadmissibly large, but we also introduce a scaling that allows us to make them satisfactorily tiny. Thus, for this scaled representation, we prove that the staircase and the $QZ$ algorithms compute the exact eigenstructure of a rational matrix $\widetilde R(λ)$ that can be expressed in exactly the same form as $R(λ)$ with the parameters defining the representation very near to those of $R(λ)$. This shows that this approach is backward stable in a structured sense.

math.NA↗

arXiv2021

Linear system matrices of rational transfer functions

In this paper we derive new sufficient conditions for a linear system matrix $$S(λ):=\left[\begin{array}{ccc} T(λ) & -U(λ) \\ V(λ) & W(λ) \end{array}\right],$$ where $T(λ)$ is assumed regular, to be strongly irreducible. In particular, we introduce the notion of strong minimality, and the corresponding conditions are shown to be sufficient for a polynomial system matrix to be strongly minimal. A strongly irreducible or minimal system matrix has the same structural elements as the rational matrix $R(λ)= W(λ) + V(λ)T(λ)^{-1}U(λ)$, which is also known as the transfer function connected to the system matrix $S(λ)$. The pole structure, zero structure and null space structure of $R(λ)$ can be then computed with the staircase algorithm and the $QZ$ algorithm applied to pencils derived from $S(λ)$. We also show how to derive a strongly minimal system matrix from an arbitrary linear system matrix by applying to it a reduction procedure, that only uses unitary equivalence transformations. This implies that numerical errors performed during the reduction procedure remain bounded. Finally, we show how to perform diagonal scalings to an arbitrary pencil such that its row and column norms are all of the order of 1. Combined with the fact that we use unitary transformation in both the reduction procedure and the computation of the eigenstructure, this guarantees that we computed the exact eigenstructure of a perturbed linear system matrix, but where the perturbation is of the order of the machine precision.

math.DS↗

arXiv2020

Block Full Rank Linearizations of Rational Matrices

Block full rank pencils introduced in [Dopico et al., Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems, Linear Algebra Appl., 2020] allow us to obtain local information about zeros that are not poles of rational matrices. In this paper we extend the structure of those block full rank pencils to construct linearizations of rational matrices that allow us to recover locally not only information about zeros but also about poles, whenever certain minimality conditions are satisfied. In addition, the notion of degree of a rational matrix will be used to determine the grade of the new block full rank linearizations as linearizations at infinity. This new family of linearizations is important as it generalizes and includes the structures appearing in most of the linearizations for rational matrices constructed in the literature. In particular, this theory will be applied to study the structure and the properties of the linearizations in [P. Lietaert et al., Automatic rational approximation and linearization of nonlinear eigenvalue problems, submitted].

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arXiv2019

Generic symmetric matrix polynomials with bounded rank and fixed odd grade

We determine the generic complete eigenstructures for $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. More precisely, we show that the set of $n \times n$ complex symmetric matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and rank at most $r$ is the union of the closures of the $\lfloor rd/2\rfloor+1$ sets of symmetric matrix polynomials having certain, explicitly described, complete eigenstructures. Then, we prove that these sets are open in the set of $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. In order to prove the previous results, we need to derive necessary and sufficient conditions for the existence of symmetric matrix polynomials with prescribed grade, rank, and complete eigenstructure, in the case where all their elementary divisors are different from each other and of degree $1$. An important remark on the results of this paper is that the generic eigenstructures identified in this work are completely different from the ones identified in previous works for unstructured and skew-symmetric matrix polynomials with bounded rank and fixed grade larger than one, because the symmetric ones include eigenvalues while the others not. This difference requires to use new techniques.

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