Congruences of a square matrix and its transpose

By Roger A. Horn, V. Sergeichuk. 2004

75 citations recorded by the source corpus.

Abstract It is known that any square matrix A over any field is congruent to its transpose: AT=STAS for some nonsingular S; moreover, S can be chosen such that S2=I, that is, S can be chosen to be involutory. We show that A and AT are ∗ congruent over any field F of characteristic not two with involution a↦ a (the involution can be the identity): A T = S T AS for some nonsingular S; moreover, S can be chosen such that S S=I , that is, S can be chosen to be coninvolutory. The short and simple proof is based on Sergeichuk's canonical form for ∗ congruence [Math. USSR, Izvestiya 31 (3) (1988) 481]. It follows that any matrix A over F can be represented as A=EB, in which E is coninvolutory and B is symmetric.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bc3625a199f9e76a0ea49d2179e30a9547f08708

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Citations in the corpus

The solution of the equation XA + AXT = 0 and its application to the theory of orbits (2011, background): “This canonical form was originally introduced by Sergeichuk in [23] for matrices over any field F with characteristic not 2 up to classification of Hermitian forms over finite extensions of F (see also the references [16, 18] by these authors).”
Canonical forms for complex matrix congruence and ∗congruence (2006, methodology): “Simpler versions of these canonical forms were given in [ 8 ], which relied on [20], and hence on the theory of representations of quivers with involution on which the latter is based.”
Canonical matrices of bilinear and sesquilinear forms (2007, methodology): “Such a simplification was given by the authors in [ 9 ] for canonical matrices of bilinear and sesquilinear forms over C; a direct proof that the matrices from [9] are canonical is given in [10, 11]; applications of these canonical matrices were obtained in [3, 4, 5, 11, 12].”
Smith Forms of Palindromic Matrix Polynomials (2011, unknown): “Our results over arbitrary fields give independent proofs of the elementary divisor pairing results presented in [25, 26] for T -palindromic pencils over the fields F = C and F = R; see also the related results in [14].”