Simultaneous Reduction of Quadratic Forms

Let x denote the real n-tuple (x1,…,xn), (xx… ) the iterated n-tuple (x1,…,xn,x1,…xn…), A = (a i j) and A\u27 a square matrix and its transpose. We wish to consider the effect of nonsingular transformations on a sum x(A1, + ... + Ak)x\u27 of k quadratic forms when the real symmetric matrices Ai and A = A, + ... + Ak are subjected to certain restrictions on their ranks. The results attained link a theorem in statistics with a theorem in ring theory (see Corollary 3). Extension of the results to hermitean forms follows easily