An Index Theorem for Monotone Matrix-Valued Functions
Werner Kratz · SIAM Journal on Matrix Analysis and Applications · 1995
The main result of this paper is the following index theorem, which is closely related to oscillation theorems on linear selfadjoint differential systems such as results by M. Morse. Let real $m \times m$-matrices $R_1 ,R_2 ,X,U$ be given, which satisfy \[ R_1 R_2^T = R_2 R_1^T ,\quad X^T U = U^T X,\quad \operatorname{rank} (R_1 ,R_2 ) = \operatorname{rank} (X^T ,U^T ) = m . \] Moreover, assume that $X(t),U(t)$ are real $m \times m$-matrix-valued functions on some interval $\mathcal{J} = [-\varepsilon ,\varepsilon ],\varepsilon > 0$, such that \[ X^T (t)U(t) = U^T (t)X(t)\quad{\text{on}}\quad\mathcal{J}, \]\[ X(t) \to X\quad {\text{and}}\quad U(t) \to U\quad{\text{as}}\quad t \to 0, \]\[ X(t)\quad{\text{is invertible for}}\quad t \in \mathcal{J}\backslash \{ 0 \},\quad {\text{and such that}} \]\[ U(t)X^{ - 1} (t)\quad {\text{is decreasing on}}\quad \mathcal{J}\backslash \{ 0 \} \] and define \[ M(t) \equiv R_1 R_2^T + R_2 U(t)X^{ - 1} (t)R_2^T ,\quad \Lambda (t) \equiv R_1 X(t) + R_2 U(t),\quad \Lambda \equiv R_1 X + R_2 U. \] Then $\operatorname{ind} M(0 + )$, $\operatorname{ind} M(0 - )$, and $\operatorname{def} \Lambda(0 + )$ exist and \[ \operatorname{ind} M(0 + ) - \operatorname{ind} M(0 - ) = \operatorname{def} \Lambda - \operatorname{def} \Lambda (0 + ) - \operatorname{def} X, \] where ind denotes the index (the number of negative eigenvalues) and def denotes the defect (the dimension of the kernel) of a matrix. The basic tool for the proof of this result consists of a theorem on the rank of a certain product of matrices, so that this rank theorem is the key result of the present paper.