Inertia theorems for pairs of matrices. (Q1430371)

Several basic matrix inertia theorems, due to Lyapunov, Taussky, Ostrowski- Schneider, Chen-Wimmer, and Loewy, are extended to pairs of real or complex matrices. In these extensions, the standard inequality \(AH_1+ H_1A^*> 0\), which means that \(AH_1+ H_1A^*\) is positive definite, or a weaker inequality \(AH_1 +H_1A^*\geq 0\), where \(A\) is a given matrix, is replaced by the inequality \(AH_1+ H_1A^*+ BH_2^* +H_2B^*> 0\), or the corresponding weaker inequality \[ AH_1+ H_1A^*+ BH^*_2+ H_2B^*\geq 0,\tag{1} \] where \((A, B)\) is a given pair of matrices. Here \(H_1\) is a Hermitian matrix (assumed to be positive definite in the Lyapunov theorem). We quote one result from the reviewed paper, a generalization of the Chen-Wimmer theorem, as a representative of the approach taken and the results obtained in the paper: Let there be given matrices \(A\in F^{p\times p }\) and \(B\in F^{p\times q}\), where \(F\) is either the real field or the complex field, and denote by \(\chi(x)\) the product of invariant factors of \([xI- A\;B]\). Let \(\pi\), \(\nu\), \(\delta\), \(\rho\) be nonnegative integers such that \(\pi+ \nu+\delta= p\). Then the following two statements are equivalent: (1) There exist a Hermitian matrix \(H_1\in F^{p\times p}\) with the inertia given by the ordered triple \((\pi,\nu,\delta)\), and a matrix \(H_2\in F^{p\times q}\) such that \[ \text{rank}[H_1\;H_2]- \text{rank\,}H_1= \rho, \] the matrix \(K\) defined by the left-hand side of (1) is positive semidefinite, and the pair \((A, [B\;K])\) is completely controllable; (2) \(\rho\leq q\), \(\rho\leq\delta\leq p\) -- \{the degree of \(\chi(x)\}\), the polynomial \(\chi(x)\) does not have roots with zero real parts, and the number of roots of \(\chi(x)\) with positive (resp. negative) real parts does not exceed \(\pi\) (resp. \(\nu\)).