Miniversal deformations of matrices of bilinear forms

Abstract

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.

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:1105.2160

References in corpus (10)

• Canonical forms for complex matrix congruence and *congruence
• Classification problems for system of forms and linear mappings
• Congruences of a square matrix and its transpose
• Canonical matrices of bilinear and sesquilinear forms
• A regularization algorithm for matrices of bilinear and sesquilinear forms
• Miniversal deformations of matrices of bilinear forms
• Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence
• Miniversal deformations of matrices under *congruence and reducing transformations
• Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils
• Generic families of matrix pencils and their bifurcation diagrams

Cited by in corpus (9)

• Miniversal deformations of matrices of bilinear forms
• Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence
• Miniversal deformations of matrices under *congruence and reducing transformations
• Neighborhood radius estimation for Arnold's miniversal deformations of complex and $p$-adic matrices
• On normal forms of complex points of small ${\mathcal{C}}^2$-perturbations of real $4$-manifolds embedded in a complex $3$-manifold
• Frobenius Degenerations of Preprojective Algebras
• A holomorphic transformation to a miniversal deformation under *congruence does not always exist
• Recovering a perturbation of a matrix polynomial from a perturbation of its linearization
• A constructive proof of Pokrzywa's theorem about perturbations of matrix pencils