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ABSTRACT: We prove Roth's type theorems for systems of matrix equations including an arbitrary mix of Sylvester and *Sylvester equations, in which also the transpose or conjugate transpose of the unknown matrices 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 × 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 Steintype matrix equations.SIAM Journal on Matrix Analysis and Applications 01/2015; · 1.81 Impact Factor 
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ABSTRACT: We construct the Hasse diagrams for the closure ordering on the sets of congruence classes of 2x2 and 3x3 matrices. In other words, we construct two directed graphs whose vertices are 2x2 or, respectively, 3x3 canonical matrices for 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. It is important to know all such matrices B if A is known only approximately. A bundle of matrices under congruence is defined as a set of square matrices A for which the pencils A+xA^T are strictly equivalent. We give motivations of this definition and construct two Hasse diagrams for the closure ordering on the sets of bundles of 2x2 and, respectively, 3x3 matrices under congruence.Linear Algebra and its Applications 12/2014; 469:305334. DOI:10.1016/j.laa.2014.11.004 · 0.98 Impact Factor 
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ABSTRACT: We study how small perturbations of a skewsymmetric matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of skewsymmetric matrix pencil orbits and bundles. In other words, we investigate when the closure of the congruence orbit (or bundle) of a skewsymmetric matrix pencil contains the congruence orbit (or bundle) of another skewsymmetric matrix pencil. The developed theory relies on our main theorem stating that a skewsymmetric matrix pencil $A\lambda B$ can be approximated by pencils strictly equivalent to a skewsymmetric matrix pencil $C\lambda D$ if and only if $A\lambda B$ can be approximated by pencils congruent to $C\lambda D$. Read More: http://epubs.siam.org/doi/abs/10.1137/140956841SIAM Journal on Matrix Analysis and Applications 11/2014; 35(4):14291443. DOI:10.1137/140956841 · 1.81 Impact Factor 
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ABSTRACT: V.I. Arnold [Russian Math. Surveys 26(2) (1971) 2943] 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 the authors in [Linear Algebra Appl. 436 (2012) 26702700].Linear Algebra and its Applications 04/2014; 446:388420. DOI:10.1016/j.laa.2014.01.016 · 0.98 Impact Factor 
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ABSTRACT: 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 blockpartitioning and / or symmetries. We study changes of the congruence canonical form of a complex skewsymmetric matrix pencil under small perturbations. The problem of computing the congruence canonical form is known to be illposed: 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 skewsymmetric matrix pencil corresponding to the first node can be transformed by an arbitrarily small perturbation to a skewsymmetric 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 skewsymmetric 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 skewsymmetric matrix pencils under congruence. Matlab functions to work with skewsymmetric 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.03/2014, Degree: Licentiate 
Article: Symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations
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ABSTRACT: The set of all solutions to the homogeneous system of matrix equations (X^TA + AX, X^TB + BX) = (0, 0), where (A,B) is a pair of symmetric matrices of the same size, is characterised. 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. Kågström, and V.V. Sergeichuk. Skewsymmetric 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 skewsymmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.The electronic journal of linear algebra ELA 01/2014; 27:118. · 0.89 Impact Factor 
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ABSTRACT: Matlab functions to work with the canonical structures for congruence and *congruence of matrices, and for congruence of symmetric and skewsymmetric matrix pencils are presented. A user can provide the canonical structure objects or create (random) matrix example setups with a desired canonical information, and compute the codimensions of the corresponding orbits: if the structural information (the canonical form) of a matrix or a matrix pencil is known it is used for the codimension computations, otherwise they are computed numerically. Some auxiliary functions are provided too. All these functions extend the Matrix Canonical Structure Toolbox. 
Article: Skewsymmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations
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ABSTRACT: The homogeneous system of matrix equations (XTA+AX,X TB+BX)=(0,0), where (A,B) is a pair of skewsymmetric 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 skewsymmetric matrix pencils.Linear Algebra and its Applications 04/2013; 438(8). DOI:10.1016/j.laa.2012.11.025 · 0.98 Impact Factor 
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ABSTRACT: We give a miniversal deformation of each pair of skewsymmetric matrices $(A,B)$ under congruence; that is, a normal form with minimal number of independent parameters to which all matrix pairs $(A+E,B+E')$ close to $(A,B)$ can be reduced by congruence transformation $ (A+E,B+E')\mapsto {\cal S}(E,E')^T (A+E,B+E') {\cal S}(E,E'), {\cal S}(0,0)=I, $ in which ${\cal S}(E,E')$ smoothly depends on the entries of $E$ and $E'$. 
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ABSTRACT: V.I. Arnold [Russian Math. Surveys 26 (2) (1971) 2943] constructed a miniversal deformation of 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 a miniversal deformation of matrices under congruence.Linear Algebra and its Applications 04/2010; DOI:10.1016/j.laa.2011.11.010 · 0.98 Impact Factor 
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ABSTRACT: Using miniversal deformations of a skewsymmetric matrix, a generalization of the Darboux theorem from symplectic geometry is derived. 
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ABSTRACT: 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 pgroups of exponent p with central commutator subgroup of order p^3 are hopeless since each of them contains • the problem of classifying symmetric bilinear mappings U × U → V , or • the problem of classifying skewsymmetric bilinear mappings U × U → V , in which U and V are vector spaces over F (consisting of p elements for pgroups (iii)) and V is 3dimensional. 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.The electronic journal of linear algebra ELA 01/2009; 18:516529. · 0.89 Impact Factor 
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ABSTRACT: 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 pgroups of exponent p with central commutator subgroup of order p 3 , are hopeless since each of them contains • the problem of classifying symmetric bilinear mappings U×U→V, or • the problem of classifying skewsymmetric bilinear mappings U×U→V, in which U and V are vector spaces over 𝔽 (consisting of p elements for pgroups, (iii)) and V is 3dimensional. 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.The electronic journal of linear algebra ELA 01/2009; · 0.89 Impact Factor

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