Contragredient equivalence: A canonical form and some applications Original Research Article

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, AT) is contragrediently equivalent to (C, CT) 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) ATA is similar to AAT;3. (3) (A, AT) is contragrediently equivalent to (AT, A);4. (4) A = Q1ATQ2 for some complex orthogonal Q1, Q2;5. (5) A = PATP for some complex orthogonal P.