Linear maps on Hermitian matrices: the stabilizer of an inertia class, II

Abstract

Abstract Let G(r s t) be the set of n-by-n Hermitian matrices with r positives negative and t zero eigenvaluesn = r + s + t. With the exception of the cases (i) r = n, (ii) s = n, and (iii) r = s t = 0 when n is even, we classify the nonsingulur linear maps T on Hermitian matrices for which T(G(r s t)) ⊆ G (r s t). Such a T is either a congruence or a congruence composed with transposition, with the additional possibility of composition with negation when r − sand t > 0. In cases (i) and (ii) above, there are definitely additional possible transformations, and a complete classification is a long standing unsolved problem. In case (iii) above, for n 4, we conjecture that the answer is congruence possibly composed with transposition and/or negation, but our methods do not cover this case. In two particular cases, (iv) r = n − 1s = 1t = 0 (n  3) and (v) r = s + 1t = 0. we show that the into assumption on T implies the nonsingularity of T, so that, in these cases, into alone implies that T is a congruence possibly composed with transposition. For n  3 we suspect that into is also sufficient for this conclusion, except that negation must also be allowed in the balanced inertia cases r = s and except for the definite inertia cases (i) and (ii).