Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils

Abstract

For a family of linear operators A( λ) : U → U over C that smoothly depend on parameters λ = (λ 1 , . . . , λ k ), V. I. Arnold obtained the simplest normal form of their matrices relative to a smoothly depending on λ change of a basis in U . We solve the same problem for a family of linear operators A( λ) : U → U over R, for a family of pairs of linear mappings A( λ) : U → V, B( λ) : U → V over C and R, and for a family of pairs of counter linear mappings A( λ) : U → V, B( λ) : V → U over C and R.This is the authors' version of a work that was published in Linear Algebra Appl. 302-303 (1999) 45-61.

Block Triangular Miniversal Deformations of Matrices and Matrix Pencils

“…the proof of Theorem 5.1 in [10], (C + P, D + Q) is a versal (respectively, miniversal) deformation of (C, D) if and only if for every pair (M, N ) of size of (C, D) there exist square matrices S and R and a pair (respectively, a unique pair) (P, Q) obtained from (P, Q) by replacing its stars with complex numbers such that…”

“…Therefore, if the entries of a matrix are known only approximately, then it is unwise to reduce it to Jordan form. Furthermore, when investigating a family of matrices smoothly depending on parameters, then although each individual matrix can be reduced to its Jordan form, it is unwise to do so since in such an operation the smoothness relative to the parameters is lost.For these reasons, Arnold [1] constructed a miniversal deformation of any Jordan canonical matrix J; that is, a family of matrices in a neighborhood of J with the minimal number of parameters, to which all matrices M close to J can be reduced by similarity transformations that smoothly depend on the entries of M (see Definition 2.1).Miniversal deformations were also constructed for: (i) the Kronecker canonical form of complex matrix pencils by Edelman, Elmroth, and Kågström [9]; another miniversal deformation (which is simple in the sense of Definition 2.2) was constructed by Garcia-Planas and Sergeichuk [10]; (ii) the Dobrovol'skaya and Ponomarev canonical form of complex contragredient matrix pencils (i.e., of matrices of counter linear operators U ⇄ V ) in [10]. Belitskii [4] proved that each Jordan canonical matrix J is permutationally similar to some matrix J # , which is called a Weyr canonical matrix and possesses the property: all matrices that commute with J # are block triangular.…”

Reduction to Versal Deformations of Matrix Pencils and Matrix Pairs with Application to Control Theory

“…If the pencil x 0 = A 0 − λB 0 is reduced to the Kronecker canonical form (this is not a restriction because of the homogeneity of the orbit), it is possible to write down explicitly the bases {c 1 [4,10].…”

“…Introduction. The Arnold technique of constructing a local canonical form, called versal deformation, of a differentiable family of square matrices under conjugation [1,2] has been generalized by several authors to matrix pencils under the strict equivalence [4,10], pairs or triples of matrices under the action of the general linear group [18], pairs of matrices under the feedback similarity [6], and triples or quadruples of matrices representing linear dynamical systems under the equivalence derived from standard transformations (the change of basis in state, input, and output spaces, state feedback, and output injection) [8,9]. Versal deformations provide a special parametrization of matrix spaces, which can be effectively applied to perturbation analysis and investigation of complicated objects like singularities and bifurcations in multiparameter dynamical systems [1,2,3,4,5,12,14,15].…”

Perturbation analysis of a matrix differential equation ẋ = ABx

“…In this paper, we describe all canonical matrix pairs (A, B) of the form (2), for which the first order induced perturbations A B + AB are nonzero for all miniversal perturbations ( A, B) = 0 in the normal form defined in [4].…”