Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils

Abstract

Abstract. 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 − 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.Key words. skew-symmetric matrix pencil, stratification, canonical structure information, orbit, bundle AMS subject classifications. 15A21, 15A22, 65F15, 47A07 DOI. 10.1137/1409568411. Introduction. How canonical information changes under perturbations, e.g., the confluence and splitting of eigenvalues of a matrix pencil, are essential issues for understanding and predicting the behavior of the physical system described by the matrix pencil. In general, these problems are known to be ill-posed: small perturbations in the input data may lead to drastical changes in the answers. The ill-posedness stems from the fact that both the canonical forms and the associated reduction transformations are discontinuous functions of the entries of A − λB. Therefore it is important to get knowledge about the canonical forms (or canonical structure information) of the pencils that are close to A − λB. One way to investigate this problem is to construct the stratification (i.e., the closure hierarchy) of orbits and bundles of the pencils [17].The stratification of matrix pencils under strict equivalence transformations [16,17,18] as well as the stratification of controllability and observability pairs [19] are known. StratiGraph [21,24] is a software tool for computing and visualization of such stratifications. The stratification of full normal rank matrix polynomials has been studied [22] and implemented in StratiGraph too (available as a prototype now).Our objective is to stratify orbits and bundles of skew-symmetric matrix pencils, i.e., A − λB with A T = −A and B T = −B, under congruence transformations. Skew-symmetric matrix pencils appear in several applications, e.g., the design of a passive velocity field controller [23], multisymplectic partial differential equations [3], and systems with bi-Hamiltonial structure [27]. Structure preserving linearizations of skew-symmetric matrix polynomials [25] allow us to investigate differential algebraic systems of higher orders via skew-symmetric matrix pencils. Canonical forms of skew-symmetric matrix pencils [29,30] and the structured staircase algorithm [2,4]

Canonical Structure Transitions of System Pencils

“…The KCF of a system pencil S in (2), given in the canonical form (12), can be obtained by substituting the block direct sum of the pencils in (12) with the direct sum of the corresponding Kronecker canonical pencils obtained as follows. Every pencil (Z q − sY q ) 11 is replaced by a strictly equivalent pencil I q − sJ q (0).…”

“…For the other types of summands it is enough just to ignore (drop) the blocking. In other words, the transformation from (12) to the KCF can be obtained by a strict equivalence transformation using permutation matrices.…”

“…Of course, all system pencils S can be transformed into KCF but not all (n + m) × (n + p) general matrix pencils can be transformed into a system pencil in the canonical form (12) with n states, m inputs, and p outputs. Nevertheless, there is only one restriction: the dimensions of the three zeroblocks in the s-matrix of (2) must be matched.…”

“…The bundle stratifications were derived for general matrix pencils [16], skewsymmetric matrix pencils and polynomials [8,12], controllability and observability pairs [17], and linearizations of polynomial matrices [11,23]. Here we present stratification of system pencil bundles.…”

An explicit description of the irreducible components of the set of matrix pencils with bounded normal rank

“…Moreover the partial containment order of orbit closures of m × n matrix pencils is also known [11], and software tools are also available to get the complete Hasse diagram of the inclusion relation between orbit closures of m × n matrix pencils [14]. The stratification of structured KCFs of structured matrix pencils or, more in general, of canonical eigenstructures of structured matrix polynomials, is currently an active area of research where many problems remain open [8,9]. In the characterization of the inclusion relation between orbit closures, the normal rank of the pencils plays a prominent role (see [11,Th.…”