Change of the *congruence canonical form of 2-by-2 matrices under perturbations

We study how small perturbations of a 2-by-2 complex matrix can change its canonical form for *congruence. We construct the Hasse diagram for the closure ordering on the set of *congruence classes of 2-by-2 matrices.Comment: 8 pages. arXiv admin note: substantial text overlap with arXiv:1105.216

Block triangular miniversal deformations of matrices and matrix pencils

For each square complex matrix, V. I. Arnold constructed a normal form with the minimal number of parameters to which a family of all matrices B that are close enough to this matrix can be reduced by similarity transformations that smoothly depend on the entries of B. Analogous normal forms were also constructed for families of complex matrix pencils by A. Edelman, E. Elmroth, and B. Kagstrom, and contragredient matrix pencils (i.e., of matrix pairs up to transformations (A,B)-->(S^{-1}AR,R^{-1}BS)) by M. I. Garcia-Planas and V. V. Sergeichuk. In this paper we give other normal forms for families of matrices, matrix pencils, and contragredient matrix pencils; our normal forms are block triangular.Comment: 14 page

An informal introduction to perturbations of matrices determined up to similarity or congruence

The reductions of a square complex matrix A to its canonical forms under transformations of similarity, congruence, or *congruence are unstable operations: these canonical forms and reduction transformations depend discontinuously on the entries of A. We survey results about their behavior under perturbations of A and about normal forms of all matrices A + E in a neighborhood of A with respect to similarity, congruence, or *congruence. These normal forms are called miniversal deformations of A; they are not uniquely determined by A + E, but they are simple and depend continuously on the entries of E

North Caucasus Baseline Project: Adygea

As of 2007, there are few signs that the Muslim community of the Adygea Republic embraces the radical Islamic tendencies seen in other parts of the North Caucasus. There is no reason to suppose that the socio-political situation in the republic is being aggravated by the Islamic revival in places such as Chechnya. Yet, there are trends that threaten to change this. The influx of Middle Eastern men, especially Muslim clerics, who visit the region on a regular basis, is a source of popular unease. Given the fact many Muslims in Adygea distrust the local clergy, the Middle Eastern missionaries working in the republic may eventually enlist support for radical Islam. The Adygh people are more likely to define themselves in terms of ethnicity than in terms of their religious affiliations. This factor mitigates possible tensions emerging from appeals by radical outsiders hoping to exploit the distrust of local clergy. The rigid social, economic, and political divisions between the Muslim and the Russian communities offer potential for future sectarian disruptions. The Nalchik violence of 2005 also led to police actions that local Muslims interpreted as persecution. Finally, although the proposal was dropped in March, 2006, the debate over a possible merger with Krasnodar sharpened differences between these groups. The threatened resignation of Adygea President Khasret Sovmen, who opposed the merger, is a factor in Islamic perceptions that they are being persecuted. On balance, Adygea represents a peaceful contrast with other republics in the North Caucasus. Separatism is not a factor and there is a general recognition that without membership in the Russian Federation Adygea wouldn’t be able to survive

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