Canonical forms for complex matrix congruence and ∗congruence

Abstract

Canonical forms for congruence and *congruence of square complex matrices were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347-353], based on Sergeichuk's paper [Math. USSR, Izvestiya 31 (3) (1988) 481-501], which employed the theory of representations of quivers with involution. We use standard methods of matrix analysis to prove directly that these forms are canonical. Our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms. We use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew-symmetric matrices as well as canonical forms for simultaneous *congruence of pairs of complex Hermitian matrices.

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

“…From the solution of this last equation we are able to recover the solution of the original equation by means of a change of variables involving the * congruency (resp., congruency) matrix leading A to C A . The canonical forms for congruence and * congruence bear a certain resemblance but they are not equal [9]. Both of them consist of three types of blocks, and these blocks coincide for just one type (Type 0), though the other two types have a similar appearance.…”

“…This suggests a natural procedure to solve (1.1), namely, to reduce A by * congruence to a simpler form and then solve the equation with this new matrix as a coefficient matrix instead of A. We will use as this simple form the canonical form for * congruence introduced by Horn and Sergeichuk [9] (see also [10,14]). …”

“…In order to recall the canonical form for * congruence, let us define the following k × k matrices as in [9],…”

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

“…Each square complex matrix is *congruent to a direct sum, uniquely determined up to permutation of summands, of matrices of the form This canonical form obtained in [11] was based on [21,Theorem 3] and was generalized to other fields in [14]. A direct proof that this form is canonical is given in [12,13].…”

“…It is not surprising that diag(λ, ±λ) and diag(µ, ν) ( λ = µ = ν = 1 and µ ≠ ±ν) have different behavior under perturbation: many properties of a nonsingular matrix A with respect to *congruence are determined by its *cosquare (A * ) −1 A (see [13,14,19]), the *cosquare of diag(λ, ±λ) has a multiple eigenvalue, and the *cosquare of diag(µ, ν) has two distinct eigenvalues.…”