Miniversal deformations of pairs of symmetric matrices under congruence

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 $(\widetilde{A},\widetilde{B})$, close to $(A,B)$ can be reduced by congruence transformation that smoothly depends on the entries of $\widetilde{A}$ and $\widetilde{B}$. 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.Comment: arXiv admin note: text overlap with arXiv:1104.249

Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence

We construct the Hasse diagrams $G_2$ and $G_3$ for the closure ordering on the sets of congruence classes of $2\times 2$ and $3\times 3$ complex matrices. In other words, we construct two directed graphs whose vertices are $2\times 2$ or, respectively, $3\times 3$ 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 $A+\lambda A^T$ belong to the same bundle under strict equivalence. In support of this definition, we show that all matrices in a congruence bundle of $2\times 2$ or $3\times 3$ matrices have the same properties with respect to perturbations. We construct the Hasse diagrams $G_2^{\rm B}$ and $G_3^{\rm B}$ for the closure ordering on the sets of congruence bundles of $2\times 2$ and, respectively, $3\times 3$ matrices. We find the isometry groups of $2\times 2$ and $3\times 3$ congruence canonical matrices.Comment: 34 page

Involving the Public in the Assessment of Community Real Estate Property

The paper argues for the need to involve the public in decision-making on abandoned community real estate property in small communities with limited financial resources. This can be achieved by giving the public the opportunity to express their opinion via a survey. For this purpose, a specific approach was developed which involves conducting a survey and evaluating the results. A particular weighting factor is given for each chosen rank of indicator. A system of 50 indicators for five different groups (interior, exterior, environment, historical and cultural value, and finance) is proposed. The indicators are divided into 38 incentives and 12 disincentives, in accordance with their impact on the final assessment of the real estate property. An example of an assessment is given and it is proposed that the survey results be categorised and analysed based on the age of respondents. The aim of this paper is to develop a way of investigating the opinion of the local community regarding abandoned municipal real estate property in the cheapest and easiest way, applicable even in small villages. Not only will this ensure the assessment is carried out, it will also involve more people in community life and increase their interest. Public participation in solving community affairs is crucial when it comes to increasing the interest of residents in the life of the territory in particular and the effective development of civil society in general. At the initial stage citizens may only engage in one-time participation; however, in the future a critical mass of caring locals will be formed who can bring forward new ideas and offer innovative solutions

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 $AX-XB=C$ has a solution if and only if the matrices $\left[\begin{matrix}A&C\\0&B\end{matrix}\right]$ and $\left[\begin{matrix}A&0\\0&B\end{matrix}\right]$ are similar. A. Dmytryshyn and B. K{\aa}gstr\"om (2015) extended Roth's criterion to systems of matrix equations $A_iX_{i'}M_i-N_iX_{i''}^{\sigma_i} B_i=C_i$ $(i=1,\dots,s)$ with unknown matrices $X_1,\dots,X_t$, 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.Comment: 11 page