### A normal form for Riemann matrices

A. A. Albert
1965 Canadian Journal of Mathematics - Journal Canadien de Mathematiques
A matrix co having p rows and 2p columns of complex number elements is called a Riemann matrix of genus p if there exists a rational 2^-rowed skew matrix C such that is positive definite Hermitian. The matrix C is then called a principal matrix of co. If co and coo are two Riemann matrices of the same genus, we say that co and coo are isomorphic if there exists a non-singular ^-rowed complex matrix a, and a non-singular 2£-rowed rational matrix A, such that (2) coo = aco^4. Then the matrix has
more » ... he property Wo Co coo' = (aoeA^A-'CiA-'YU^W = a(uC<a')a' = 0, and To = ico 0 Coco' = i(aco^)[^-1 C(^-1 ) / M^/â' = ia(uCû f )â' = ayâ' is positive definite Hermitian when y is positive definite Hermitian. It follows that, if (2) holds, then co is a Riemann matrix with C as principal matrix if and only if coo is a Riemann matrix with Co as principal matrix. The relation of isomorphism for Riemann matrices is readily seen to be an equivalence relation. Two Riemann matrices co and coo are said to be equivalent if (2) holds with A a unimodular integral matrix. The relation of equivalence is also easily seen to be an equivalence relation. It is also clear that, if C is a principal matrix of co, and t is any positive rational number, the matrix tC is a principal matrix of co. Then every Riemann matrix co has a principal matrix C whose elements are all integers. Indeed C may be selected so that the greatest common divisor of its (integral) elements is 1.