The equivalence of pairs of Hermitian matrices

Abstract

Two pairs of n-ary Hermitian forms with nXn matrices A, B and C, D with elements in the complex field are equivalent if there exists a nonsingular matrix T such that T'A T = C and T'BT = D, where T' is the conjugate-transpose of T. As is usual in the study of equivalence of pairs of matrices the work divides itself into the consideration of the non-singular and singular cases. These two cases are taken up in Parts II and I respectively. In the non-singular case the rank of pA +a-B is n except for special values of p and o-. It has frequently been pointed out that in this case no generality is lost by assuming B is of rank n. In the singular case the rank r of pA +?-B is less than n for all values of p and a-, but as above no generality is lost in assuming that the rank r of B is the maximum rank of pA +?-B. By the elementary divisors of a pair of matrices A, B is meant the elementary divisors of A -NB when B is non-singular, and the elementary divisors of pA +a-B when B is singular but the determinant IpA+a-B is not identically zero in p and o-. In the non-singular case, the well known necessary and sufficient condition for the equivalence in any field of pairs of bilinear forms, or of their corresponding matrices, and for the equivalence in the field of complex numbers of pairs of symmetric matrices is that the pairs have the same elementary divisors. This condition is known to be not sufficient