An alternative theorem for quadratic forms and extensions
This note concerns variations on the following theorem of \textit{Y. Yuan} [Math. Program., Ser., A 47, No. 1, 53-63 (1990; Zbl 0711.90062)]. Theorem: For two real symmetric matrices \(A_1\) and \(A_2\) of order \(n\) by \(n\) the following statements are equivalent: \[ \max \bigl\{ \langle x, A_1x \rangle, \langle x, A_2x \rangle \bigr\} \geq 0 \text{ for all } x \in\mathbb{R}^n; \tag{a} \] (b) there exist \(t_1 \in\mathbb{R}_+\) and \(t_2 \in\mathbb{R}_+\) with \(t_1 + t_2 = 1\), such that the matrix is positive semidefinite. Different extensions of this theorem are discussed, but it is also noticed that the theorem does not extend to the case of more than two quadratic forms. The clearly arranged proof uses convex geometry in arbitrary real topological linear spaces and helps to understand the geometric aspect hidden in the above theorem. Potential applications of the above theorem in other areas are announced but not explored in detail.
scientific article; zbMATH DE number 740257