Computing the inertias in symmetric matrix pencils (Q1330008)

Let \(S\) and \(T\) be \(n \times n\) real symmetric matrices and define \(P(S,T) = \{aS + bT : a,b \in \mathbb{R}\}\). A generalized pencil eigenvalue of the pencil \(P(S,T)\) is a pair \((a,b) \neq (0,0)\) such that \(aS + bT\) is singular. Suppose \(S\) is nonsingular and consider the open sectors of \(\mathbb{R}^ 2\) determined by the lines through the origin whose slopes are the eigenvalues of \(S^{-1}T\). The author studies what happens to the inertia of \(aS + bT\) as \((a,b)\) traverses a circle. It changes exactly when \((a,b)\) moves from one sector to an adjacent one; if there is a positive definite matrix in the pencil then the inertia changes monotonically; and if every element of the pencil except zero is indefinite and \(S^{-1}T\) has \(n\) distinct real eigenvalues, then the inertia cannot change monotonically as \((a,b)\) traverses any semicircle. This leads to an open question: for which \((S,T)\) is the inertia monotonic on some semicircle? An algorithm for determining the inertia of a pencil and several examples are included.