On the congruence of square real matrices Original Research Article

We show that if A and B are real n by n matrices which are *-congruent (i.e., P*AP=B for some invertible complex matrix P), then A and B are congruent over the real numbers (i.e., QTAQ=B for some invertible real matrix Q). This statement remains true if P is assumed to be an invertible quaternionic matrix.