CONTRAGREDIENT EQUIVALENCE - A CANONICAL FORM AND SOME APPLICATIONS

Let A and C be m-by-n complex matrices, and let B and D be n-by-m comp lex matrices. The pair (A, B) is cantragrediently equivalent to the pa ir (C, D) if there are square nonsingular complex matrices X and Y suc h that XAY(-1)=C and YBX(-1)=D. Contragredient equivalence is a common generalization of four basic equivalence relations: similarity, consi milarity, complex orthogonal equivalence, and unitary equivalence. We develop a complete set of invariants and an explicit canonical form fo r contragredient equivalence and show that (A, A(T)) is contragedientl y equivalent to (C, C-T) if and only if there are complex orthogonal m atrices P and Q such that C=PAQ. Using this result, we show that the f ollowing are equivalent for a given n-by-n complex matrix A: (1) A=QS for some complex orthogonal Q and some complex symmetric S; (2) A(T)A is similar to AA(T); (3) (A, A(T)) is contragrediently equivalent to ( A(T), A); (4) A = Q(1)A(T)Q(2) for some complex orthogonal Q(1), Q(2); (5) A = PA(T)P for some complex orthogonal P. We then consider a line ar operator phi on n-by-n complex matrices that shares the following p roperties with transpose operators: for every pair of n-by-n complex m atrices A and B, (a) phi preserves the spectrum of A, (b) phi(phi(A)) = A, and (c) phi(AB) = phi(B)phi(A). We show that (A, phi(A)) is contr agrediently similar to (B, phi(B)) if and only if A = X(1)BX(2) for so me nonsingular X(1), X(2) that satisfy X(1)(-1) = phi(X(1)) and X(2)(- 1) = phi(X(2)). We also consider a factorization of the form A = XY, w here X(-1) = phi(X) and Y = phi(Y). We use the canonical form for the contragredient equivalence relation to give a new proof of a theorem o f Flanders concerning the relative sizes of the nilpotent Jordan block s 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.