A pair of matrices sharing common Lyapunov solutions--A closer look
The inertia of a complex \(n\times n\)-matrix \(A\) is the triple \((\nu ,\delta ,\pi)\) of the numbers of eigenvalues of \(A\) with negative, zero and positive real parts (one has \(\nu +\delta +\pi =n\)). The inertia is regular whenever \(\delta =0\). For a pair of matrices \(A\), \(B\) with regular inertia if \(HA+A^*H\) and \(HB+B^*H\) are both positive definite for some Hermitian matrix \(H\), then all matrices from the convex hull conv\((A,A^{-1},B,B^{-1})\) have identical regular inertia. This implies that both conv\((A,B)\) and conv\((A,B^{-1})\) consist of non-singular matrices. The converse implications, in general, are not true. The authors seek situations where these implications do hold, in particular, in the case when \(A\), \(B\) are real \(2\times 2\)-matrices. They discuss also some aspects of these statements in the case of \(n\times n\)-matrices and they introduce a connection to the characterization of the convex hull of matrices with regular inertia. The differences between the real and complex cases are indicated. For a matrix \(A\) with regular inertia denote by \({\mathcal H}(A)\) the set of all solutions to the Lyapunov inclusion associated with \(A\), i.e. the set of Hermitian matrices \(H\) for which the matrix \(HA+A^*H\) is a positive definite Hermitian matrix. The formulations of the results of the paper include geometric properties of the sets \({\mathcal H}(A)\).
convex sets of matrices with regular inertia
scientific article; zbMATH DE number 1880029