The equation XA+AX*=0 and the dimension of *congruence orbits

Terán, Fernando De; Dopico, Froilán M.

doi:10.13001/1081-3810.1449

Cited by 19 publications

(29 citation statements)

References 15 publications

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“…Some steps toward the understanding of stratifications of matrix pencils with mixed types of symmetries have been done recently, e.g., miniversal deformations [10,11] and codimension computations [6,7]; the stratifications of 2 × 2 and 3 × 3 matrices of bilinear forms which give the stratifications of 2 × 2 and 3 × 3 symmetric/skewsymmetric matrix pencils are given in [12]. For matrix pencils with two symmetric matrices see also [9,15].…”

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confidence: 99%

Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils

Dmytryshyn

Kågström

2014

SIAM J. Matrix Anal. & Appl.

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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]

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confidence: 99%

Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils

Dmytryshyn

Kågström

2014

SIAM J. Matrix Anal. & Appl.

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“…The term "*congruence orbit" is often used instead of "*congruence class" (see De Terán and Dopico [2]). The problem that we consider can be called "the stratification of orbits of matrices under *congruence" by analogy with the stratification of orbits of matrices under similarity and of matrix pencils [7,8,15].…”

Section: Elamentioning

confidence: 99%

“…Note that the codimensions of congruence and *congruence classes were calculated in [1,5] and [2,6], respectively. By [22, Part III, Theorem 1.7], the boundary of each *congruence class is a union of *congruence classes of strictly lower dimension, which ensures the following lemma.…”

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confidence: 99%

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

Futorny¹,

Klimenko²,

Sergeichuk³

2014

ELA

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“…AX + X ⋆ A = 0 has been considered in [9,10], where the authors present a (non-numerical) method to find the set of solutions of (1.3) through the use of the canonical form of the matrix A under ⋆-congruence [20]. References [9,10] pay special attention to the relationship between (1.3) and the orbit of A under the action of ⋆-congruence.…”

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confidence: 99%

“…References [9,10] pay special attention to the relationship between (1.3) and the orbit of A under the action of ⋆-congruence. More precisely, the dimension of the solution space of (1.3) is shown to be equal to the codimension of this orbit.…”

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Consistency and efficient solution of the Sylvester equation for *-congruence

Terán¹,

Dopico²

2011

ELA

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Abstract. In this paper, the matrix equation AX + X ⋆ B = C is considered, where the matrices A and B have sizes m × n and n × m, respectively, the size of the unknown X is n × m, and the operator (·) ⋆ denotes either the transpose or the conjugate transpose of a matrix. In the first part of the paper, necessary and sufficient conditions for the existence and uniqueness of solutions are reviewed. Algorithms, 51:209-238, 2009.]. This review generalizes to fields of characteristic different from two the existence condition that Wimmer originally proved for the complex field. In the second part, an algorithm is developed, in the real or complex square case m = n, to solve the equation in O(n 3 ) flops when the solution is unique. This algorithm is based on the generalized Schur decomposition of the matrix pencil A − λB ⋆ . The equation AX + X ⋆ B = C is connected with palindromic eigenvalue problems and, as a consequence, the square complex case has attracted recently the attention of several authors.

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The equation XA+AX=0 and the dimension of congruence orbits

Cited by 19 publications

References 15 publications

Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils

Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils

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

Consistency and efficient solution of the Sylvester equation for *-congruence

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