40 research outputs found
Miniversal deformations of pairs of symmetric matrices under congruence
For each pair of complex symmetric matrices we provide a normal form
with a minimal number of independent parameters, to which all pairs of complex
symmetric matrices , close to can be
reduced by congruence transformation that smoothly depends on the entries of
and . Such a normal form is called a miniversal
deformation of 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 to
its miniversal deformation.Comment: arXiv admin note: text overlap with arXiv:1104.249
Schur decomposition of several matrices
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 the other words, we describe when the Schur form of a collection of
matrices exists and how to find it.Comment: 10 page
Miniversal deformations of matrices of bilinear forms
V.I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] 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.Comment: 39 pages. The first version of this paper was published as Preprint
RT-MAT 2007-04, Universidade de Sao Paulo, 2007, 34 p. The work was done
while the second author was visiting the University of Sao Paulo supported by
the Fapesp grants (05/59407-6 and 2010/07278-6). arXiv admin note:
substantial text overlap with arXiv:1105.216
Generalization of Roth's solvability criteria to systems of matrix equations
W.E. Roth (1952) proved that the matrix equation has a solution if
and only if the matrices and
are similar. A. Dmytryshyn
and B. K{\aa}gstr\"om (2015) extended Roth's criterion to systems of matrix
equations with
unknown matrices , in which every is , , or
. 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.Comment: 11 page
The Dynamical Functional Particle Method for Multi-Term Linear Matrix Equations
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 dynamical 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 Hermitian positive definite or negative definite coefficients. In numerical experiments, our MATLAB 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