An effective criterion for congruence of real symmetric matrix pairs

AbstractThis paper gives a rational method of determining the congruence of m × m real symmetric pairs over the reals R. If (S1,T1) and (S2,T2) are nonsingular pairs, then (S1,T1) is congruent to (S2,T2) over R if and only if S1−1T1 is similar to S2−1T2 and the signatures of S1ƒ(S1−1T1)k g(S1−1T1) and S2ƒ(S2−1T2)kg (S2−1T2) are equal for k =0,1,2,…,m−1 and for all g(x) in P, where P is a relatively small set of real polynomials and ƒ(x) is a fixed polynomial. This result is then extended to singular pairs using theorems on minimal indices