Trace minimization and definiteness of symmetric pencils (Q1805221)

The authors are interested in the following extremal problem for a general pair of symmetric matrices \(A,B\): \(TrX^ T AX = \min\), \(X^ T BX = J_ 1\) where \(B\) is nonsingular, \(J = \text{diag} (I_{p_ 1}, - I_{q_ 1})\), and \(p_ 1 \leq p\), \(q_ 1 \leq q\), with \((p,q)\) as the inertia of \(B\). They first prove an interlacing theorem of Cauchy type for a pair \(A,B\) with \[ {B = J = \text{diag} (\varepsilon_ 1, \dots, \varepsilon_ n)},\quad\varepsilon_ i \in \{- 1,1\}. \] The theorem ascertains the interlacing relation of the eigenvalues \(\theta^ +_ i\), \(\theta^ - _ j\) of the pairs \(A,J\) and \(H,J\) respectively, where \[ A = \left( \begin{smallmatrix} H & K^ T \\ K & U \end{smallmatrix} \right) \quad J = \left( \begin{smallmatrix} J_ 1 & O \\ O & J_ 2 \end{smallmatrix} \right). \] They next show the existence of the minimum listed above and that this existence implies the positive-definiteness of the pair \(A,B\) and that any minimizing \(X_ 1\) spans a spectral subspace for the pair \(A,J\).