Congruence of symmetric matrices

Abstract

In this note we consider necessary and sufficient conditions over a ring K in order to obtain the diagonalization by congruence of symmetric matrices over K. For instance, we show: (1) K is a field of characteristic different from 2 with K= K ² if and only if K is a ring and the rank of matrices characterizes the class of congruency; (2) If K is an ordered field and K ² is the subset of positives, two symmetric matrices are congruent if and only if they have the same rank and signature. More precisely, K is an ordered field and K ² is the subset of positives if and only if K is a ring, K ² is closed for the sum and for every symmetric matrix A over K there exists a unique s ??N such that A is congruent to

diag(1, ..^s.,1,−1,.^r−s..,−1,0,.^n−r..,0)

where r= rank(/1). Finally, we expose a wide family of not algebraically closed fields K satisfying the conditions in 1), and not real closed fields K satisfying 2).