Geodesic
On a Note by H. Schwerdtfeger
Canadian mathematical bulletin, Tome 1 (1958) no. 3, pp. 181-182

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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 formy' 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: (i) A is of the congruence type [+, -,..., -]. (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. 
Scherk, Peter. On a Note by H. Schwerdtfeger. Canadian mathematical bulletin, Tome 1 (1958) no. 3, pp. 181-182. doi: 10.4153/CMB-1958-020-9
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