Let F be a field of characteristic different from 2. It is shown that the problems of classifying
(i) local commutative associative algebras over F with zero cube radical,
(ii) Lie algebras over F with central commutator subalgebra of dimension 3, and
(iii) finite p-groups of exponent p with central commutator subgroup of order are hopeless since each of them contains
• the problem of classifying symmetric bilinear mappings UxU → V , or
• the problem of classifying skew-symmetric bilinear mappings UxU → V ,
in which U and V are vector spaces over F (consisting of p elements for p-groups (iii)) and V is 3-dimensional. The latter two problems are hopeless since they are wild; i.e., each of them contains the problem of classifying pairs of matrices over F up to similarity.
We show that the set of n x n complex symmetric matrix pencils of rank at most r is the union of the closures of left perpendicular r/2 Right perpendicular + 1 sets of matrix pencils with some, explicitly described, complete eigenstructures. As a consequence, these are the generic complete eigenstructures of n x n complex symmetric matrix pencils of rank at most r. We also show that the irreducible components of the set of n x n symmetric matrix pencils with rank at most r, when considered as an algebraic set, are among these closures.
We show that the set of m x 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 x 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 [24].
We determine the generic complete eigenstructures for n x 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 left perpendicularrd/2right perpendicular + 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 x 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 1, because the symmetric ones include eigenvalues while the others not. This difference requires using new techniques.
We obtain the generic complete eigenstructures of complex Hermitian n x n matrix pencils with rank at most r (with r <= 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 x n pencils with the same complete eigenstructure (up to the specific values of the distinct 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 nonreal eigenvalues. Moreover, we will see that the sign characteristic of the real eigenvalues plays a relevant role for determining the generic eigenstructures.
Miniversal deformations for pairs of skew-symmetric matrices under congruence are constructed. To be precise, for each such a pair (A, B) we provide a normal form with a minimal number of independent parameters to which all pairs of skew-symmetric matrices ((A) over tilde (,) (B) over tilde), close to (A, B) can be reduced by congruence transformation which smoothly depends on the entries of the matrices in the pair ((A) over tilde (,) (B) over tilde). An upper bound on the distance from such a miniversal deformation to (A, B) is derived too. We also present an example of using miniversal deformations for analyzing changes in the canonical structure information (i.e. eigenvalues and minimal indices) of skew-symmetric matrix pairs under perturbations.
For each pair of complex symmetric matrices (A, B) we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices ((A) over tilde (B) over tilde), close to (A, B) can be reduced by congruence transformation that smoothly depends on the entries of (A ) over tilde and (B) over tilde. Such a normal form is called a miniversal deformation of (A, B) under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from (A, B) to its miniversal deformation.
A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory, that relates perturbations in the linearization to equivalent perturbations in the corresponding matrix polynomial, is needed. In this paper we develop an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. Moreover we find transformation matrices that, via strict equivalence, transform a perturbation of the linearization to the linearization of a perturbed polynomial. For simplicity, we present the results for the first companion linearization but they can be generalized to a broader class of linearizations.
Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it.
Investigating the properties, explaining, and predicting the behaviour of a physical system described by a system (matrix) pencil often require the understanding of how canonical structure information of the system pencil may change, e.g., how eigenvalues coalesce or split apart, due to perturbations in the matrix pencil elements. Often these system pencils have different block-partitioning and / or symmetries. We study changes of the congruence canonical form of a complex skew-symmetric matrix pencil under small perturbations. The problem of computing the congruence canonical form is known to be ill-posed: both the canonical form and the reduction transformation depend discontinuously on the entries of a pencil. Thus it is important to know the canonical forms of all such pencils that are close to the investigated pencil. One way to investigate this problem is to construct the stratification of orbits and bundles of the pencils. To be precise, for any problem dimension we construct the closure hierarchy graph for congruence orbits or bundles. Each node (vertex) of the graph represents an orbit (or a bundle) and each edge represents the cover/closure relation. Such a relation means that there is a path from one node to another node if and only if a skew-symmetric matrix pencil corresponding to the first node can be transformed by an arbitrarily small perturbation to a skew-symmetric matrix pencil corresponding to the second node. From the graph it is straightforward to identify more degenerate and more generic nearby canonical structures. A necessary (but not sufficient) condition for one orbit being in the closure of another is that the first orbit has larger codimension than the second one. Therefore we compute the codimensions of the congruence orbits (or bundles). It is done via the solutions of an associated homogeneous system of matrix equations. The complete stratification is done by proving the relation between equivalence and congruence for the skew-symmetric matrix pencils. This relation allows us to use the known result about the stratifications of general matrix pencils (under strict equivalence) in order to stratify skew-symmetric matrix pencils under congruence. Matlab functions to work with skew-symmetric matrix pencils and a number of other types of symmetries for matrices and matrix pencils are developed and included in the Matrix Canonical Structure (MCS) Toolbox.
We study how elementary divisors and minimal indices of a skew-symmetric matrix polynomial of odd degree may change under small perturbations of the matrix coefficients. We investigate these changes qualitatively by constructing the stratifications (closure hierarchy graphs) of orbits and bundles for skew-symmetric linearizations. We also derive the necessary and sufficient conditions for the existence of a skew-symmetric matrix polynomial with prescribed degree, elementary divisors, and minimal indices.
Developing theory, algorithms, and software tools for analyzing matrix pencils whose matrices have various structures are contemporary research problems. Such matrices are often coming from discretizations of systems of differential-algebraic equations. Therefore preserving the structures in the simulations as well as during the analyses of the mathematical models typically means respecting their physical meanings and may be crucial for the applications. This leads to a fast development of structure-preserving methods in numerical linear algebra along with a growing demand for new theories and tools for the analysis of structured matrix pencils, and in particular, an exploration of their behaviour under perturbations. In many cases, the dynamics and characteristics of the underlying physical system are defined by the canonical structure information, i.e. eigenvalues, their multiplicities and Jordan blocks, as well as left and right minimal indices of the associated matrix pencil. Computing canonical structure information is, nevertheless, an ill-posed problem in the sense that small perturbations in the matrices may drastically change the computed information. One approach to investigate such problems is to use the stratification theory for structured matrix pencils. The development of the theory includes constructing stratification (closure hierarchy) graphs of orbits (and bundles) that provide qualitative information for a deeper understanding of how the characteristics of underlying physical systems can change under small perturbations. In turn, for a given system the stratification graphs provide the possibility to identify more degenerate and more generic nearby systems that may lead to a better system design.
We develop the stratification theory for Fiedler linearizations of general matrix polynomials, skew-symmetric matrix pencils and matrix polynomial linearizations, and system pencils associated with generalized state-space systems. The novel contributions also include theory and software for computing codimensions, various versal deformations, properties of matrix pencils and matrix polynomials, and general solutions of matrix equations. In particular, the need of solving matrix equations motivated the investigation of the existence of a solution, advancing into a general result on consistency of systems of coupled Sylvester-type matrix equations and blockdiagonalizations of the associated matrices.
The set POLd,rm×n of m×n complex matrix polynomials of grade d and (normal) rank at most r in a complex (d+1)mn dimensional space is studied. For r=1,...,min{m,n}−1, we show that POLd,rm×n is the union of the closures of the rd+1 sets of matrix polynomials with rank r, degree exactly d, and explicitly described complete eigenstructures. In addition, for the full-rank rectangular polynomials, i.e. r=min{m,n} and m≠n, we show that POLd,rm×n coincides with the closure of a single set of the polynomials with rank r, degree exactly d, and the described complete eigenstructure. These complete eigenstructures correspond to generic m×n matrix polynomials of grade d and rank at most r.
We show that the set of m×m complex skew-symmetric matrix polynomials of odd 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 the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic m×m complex skew-symmetric matrix polynomials of odd grade d and rank at most 2r. In particular, this result includes the case of skew-symmetric matrix pencils (d=1).
Recent years have seen a renewal of interest in multi-term linear matrix equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dy-namical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all linear matrix equations with Hermi-tian positive definite or negative definite coefficients. In numerical experiments, our MAT -LAB implementation outperforms existing methods for the solution of multi-term Sylvester equations. For the Sylvester equation AX + XB = C, in particular, it can be faster and more accurate than the built-in implementation of the Bartels-Stewart algorithm, when A and B are well conditioned and have very different size.
We classify pairs of linear mappings (U -> V, U/U' -> V') in which U, V are finite dimensional vector spaces over a field IF, and U', are their subspaces. (C) 2016 Elsevier Inc. All rights reserved.
W.E. Roth (1952) proved that the matrix equation AX - XB = C has a solution if and only if the matrices [Graphics] and [Graphics] are similar. A. Dmytryshyn and B. Kagstrom (2015) extended Roth's criterion to systems of matrix equations A(i)X(i')M(i) - (NiXi"Bi)-B-sigma i = Ci (i = 1,..., s) with unknown matrices X1,, X-t, in which every X-sigma is X, X-T, or X*. We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations. (C) 2017 Elsevier Inc. All rights reserved.
We construct the Hasse diagrams G2 and G3 for the closure ordering on the sets of congruence classes of 2 × 2 and 3 × 3 complex matrices. In other words, we construct two directed graphs whose vertices are 2 × 2 or, respectively, 3 × 3 canonical matrices under congruence, and there is a directed path from A to B if and only if A can be transformed by an arbitrarily small perturbation to a matrix that is congruent to B. A bundle of matrices under congruence is defined as a set of square matrices A for which the pencils A + λAT belong to the same bundle under strict equivalence. In support of this definition, we show that all matrices in a congruence bundle of 2 × 2 or 3 × 3 matrices have the same properties with respect to perturbations. We construct the Hasse diagrams G2 B and G3 B for the closure ordering on the sets of congruence bundles of 2 × 2 and, respectively, 3 × 3 matrices. We find the isometry groups of 2 × 2 and 3 × 3 congruence canonical matrices.
Arnold [V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26 (2) (1971) 29–43] constructed miniversal deformations of square complex matrices under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We construct miniversal deformations of matrices under congruence.
Arnold (1971) [1] constructed a miniversal deformation of a square complex matrix under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We give miniversal deformations of matrices of sesquilinear forms; that is, of square complex matrices under *congruence, and construct an analytic reducing transformation to a miniversal deformation. Analogous results for matrices under congruence were obtained by Dmytryshyn, Futorny, and Sergeichuk (2012) [11].
We investigate the changes under small perturbations of the canonical structure information for a system pencil (A B C D) − s (E 0 0 0), det(E) ≠ 0, associated with a (generalized) linear time-invariant state-space system. The equivalence class of the pencil is taken with respect to feedback-injection equivalence transformation. The results allow to track possible changes under small perturbations of important linear system characteristics.
We investigate the changes of the canonical structure information under small perturbations for a system pencil associated with a (generalized) linear time-invariant state-space system. The equivalence class of the pencil is taken with respect to feedback-injection equivalence transformations. The results allow us to track possible changes of important linear system characteristics under small perturbations.
Matlab functions to work with the canonical structures for congru-ence and *congruence of matrices, and for congruence of symmetricand skew-symmetric matrix pencils are presented. A user can providethe canonical structure objects or create (random) matrix examplesetups with a desired canonical information, and compute the codi-mensions of the corresponding orbits: if the structural information(the canonical form) of a matrix or a matrix pencil is known it isused for the codimension computations, otherwise they are computednumerically. Some auxiliary functions are provided too. All thesefunctions extend the Matrix Canonical Structure Toolbox.
We study how small perturbations of general matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy (stratification) graphs of matrix polynomials’ orbits and bundles. To solve this problem, we construct the stratification graphs for the first companion Fiedler linearization of matrix polynomials. Recall that the first companion Fiedler linearization as well as all the Fiedler linearizations is matrix pencils with particular block structures. Moreover, we show that the stratification graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler linearizations have the same geometry (topology). This geometry coincides with the geometry of the space of matrix polynomials. The novel results are illustrated by examples using the software tool StratiGraph extended with associated new functionality.
We study how small perturbations of matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy graphs (stratifications) of orbits and bundles of matrix polynomial Fiedler linearizations. We show that the stratifica-tion graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler lineariza-tions have the same geometry (topology). The results are illustrated by examples using the software tool StratiGraph.
We prove Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and $\star$-Sylvester equations, in which the transpose or conjugate transpose of the unknown matrices also appear. In full generality, we derive consistency conditions by proving that such a system has a solution if and only if the associated set of $2 \times 2$ block matrix representations of the equations are block diagonalizable by (linked) equivalence transformations. Various applications leading to several particular cases have already been investigated in the literature, some recently and some long ago. Solvability of these cases follow immediately from our general consistency theory. We also show how to apply our main result to systems of Stein-type matrix equations.
We study how small perturbations of a skew-symmetric matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of skew-symmetric matrix pencil orbits and bundles. In other words, we investigate when the closure of the congruence orbit (or bundle) of a skew-symmetric matrix pencil contains the congruence orbit (or bundle) of another skew-symmetric matrix pencil. The developed theory relies on our main theorem stating that a skew-symmetric matrix pencil A - lambda B can be approximated by pencils strictly equivalent to a skew-symmetric matrix pencil C - lambda D if and only if A - lambda B can be approximated by pencils congruent to C - lambda D.
We study how small perturbations of a skew-symmetric matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of skew-symmetric matrix pencil orbits and bundles. In other words, we investigate when the closure of the congruence orbit (or bundle) of a skew-symmetric matrix pencil contains the congruence orbit (or bundle) of another skew-symmetric matrix pencil. This theory relies on our main theorem stating that a skew-symmetric matrix pencil A-λB can be approximated by pencils strictly equivalent to a skew-symmetric matrix pencil C-λD if and only if A-λB can be approximated by pencils congruent to C-λD.
The homogeneous system of matrix equations (X(T)A + AX, (XB)-B-T + BX) = (0, 0), where (A, B) is a pair of skew-symmetric matrices of the same size is considered: we establish the general solution and calculate the codimension of the orbit of (A, B) under congruence. These results will be useful in the development of the stratification theory for orbits of skew-symmetric matrix pencils.
The set of all solutions to the homogeneous system of matrix equations (X-T A + AX, X-T B + BX) = (0, 0), where (A, B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A, B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. Kagstrom, and V. V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375-3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.
Machine learning models are now widely used in a variety of tasks. However, they are vulnerable to adversarial perturbations. These are slight, intentionally worst-case, modifications to input that change the model’s prediction with high confidence, without causing a human eye to spot a difference from real samples. The detection of adversarial samples is an open problem. In this work, we explore a novel method towards adversarial image detection with linear algebra approach. This method is built on a comparison of distances to the centroids for a given point and its neighbors. The method of adversarial examples detection is explained theoretically, and the numerical experiments are done to illustrate the approach.
Motion capture (MoCap) technology is an essential tool for recording and analyzing movements of objects or humans. However, MoCap systems frequently encounter the challenge of missing data, stemming from mismatched markers, occlusion, or equipment limitations. Recovery of these missing data is imperative to maintain the reliability and integrity of MoCap recordings. This paper introduces a novel application of the tensor framework for MoCap data completion. We propose three completion algorithms based on the canonical polyadic (CP) decomposition of tensors. The first algorithm utilizes CP decomposition to capture the low-rank structure of the tensor. However, relying only on low-rank assumptions may be insufficient to deal with complex motion data. Thus, we propose two modified CP decompositions that incorporate additional information, SmoothCP and SparseCP decompositions. SmoothCP integrates piecewise smoothness prior, while SparseCP incorporates sparsity prior, each aiming to improve the accuracy and robustness of MoCap data recovery. To compare and evaluate the merit of the proposed algorithms over other tensor completion methods in terms of several evaluation metrics, we conduct numerical experiments with different MoCap sequences from the CMU motion capture dataset.
The Gaussian phase-space representation can be used to implement quantum dynamics for fermionic particles numerically. To improve numerical results, we explore the use of dynamical diffusion gauges in such implementations. This is achieved by benchmarking quantum dynamics of few-body systems against independent exact solutions. A diffusion gauge is implemented here as a so-called noise-matrix, which satisfies a matrix equation defined by the corresponding Fokker-Planck equation of the phase-space representation. For the physical systems with fermionic particles considered here, the numerical evaluation of the new diffusion gauges allows us to double the practical simulation time, compared with hitherto known analytic noise-matrices. This development may have far reaching consequences for future quantum dynamical simulations of many-body systems.
In this paper, we consider the solvability conditions of some Sylvester-type quaternion matrix equations. We establish some practical necessary and sufficient conditions for the existence of solutions of a Sylvester-type quaternion matrix equation with five unknowns through the corresponding equivalence relations of the block matrices. Moreover, we present some solvability conditions to some Sylvester-type quaternion matrix equations, including those involving Hermicity. The findings of this article extend related known results.