Contragredient equivalence: A canonical form and some applications

Abstract

Let A and C be m-by- n complex matrices, and let B and D be n-by- m complex matrices. The pair ( A, B) is contragrediently equivalent to the pair ( C, D) if there are square nonsingular complex matrices X and Y such that XAY −1 = C and YBX −1 = D. Contragredient equivalence is a common generalization of four basic equivalence relations: similarity, consimilarity, complex orthogonal equivalence, and unitary equivalence. We develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that ( A, A T ) is contragrediently equivalent to ( C, C T ) if and only if there are complex orthogonal matrices P and Q such that C = PAQ. Using this result, we show that the following are equivalent for a given n-by- n complex matrix A: 1. (1) A = QS for some complex orthogonal Q and some complex symmetric S; 2. (2) A T A is similar to AA T ; 3. (3) ( A, A T ) is contragrediently equivalent to ( A T , A); 4. (4) A = Q 1 A T Q 2 for some complex orthogonal Q 1, Q 2; 5. (5) A = PA T P for some complex orthogonal P. We then consider a linear operator φ on n-by- n complex matrices that shares the following properties with transpose operators: for every pair of n-by- n complex matrices A and B, (a) φ. preserves the spectrum of A, (b) φ( φ( A)) = A, and (c) φ( AB) = φ( B) φ( A). We show that ( A, φ( A)) is contragrediently similar to ( B, φ( B)) if and only if A = X 1 BX 2 for some nonsingular X 1, X 2 that satisfy X −1 1 = φ( X 1) and X −1 2 = φ( X 2). We also consider a factorization of the form A = XY, where X −1 = φ( X) and Y = φ( Y). We use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of Flanders concerning the relative sizes of the nilpotent Jordan blocks of AB and BA. We present a sufficient condition for the existence of square roots of AB and BA and close with a canonical form for complex orthogonal equivalence.