On a Note by H. Schwerdtfeger

Extract

Let x, y denote real column vectors with n elements. Let A be a regular symmetric real n×n matrix. Dashes indicate transposition.

If x is fixed, x' Ax > 0, the discriminant of A at x is the quadratic form

y' where S= S(x)= x'x'Ax ¨ Axx'A.

In Can. Math. Bull. 1, pp. 175-179, Dr. Schwerdtfeger proved the equivalence of the following properties of A:

1. (i) A is of the congruence type [+, -,..., -].

2. (ii) y' Sy≤ 0 for all y, equality holding if and only if y is a multiple of x. His note is of particular interest because he also discusses the eigen-values of S. If only the quoted result is aimed at, the following procedure may be shorter.