A pair of matrices sharing common Lyapunov solutions - A closer look

Let A,B be a pair of matrices with regular inertia. If HA+A ^*H and HB+B ^*H are both positive definite for some Hermitian matrix H then all matrices in conv(A,A ^-1,B,B ^-1) have identical regular inertia. This, in turn, implies that both conv(A,B) and conv(A,B ^-1) consist of non-singular matrices. In general, neither of the converse implications holds. In this paper we seek situations where they do hold, in particular, when A and B are real 2×2 matrices. Several aspects of the above statements for n×n matrices are discussed. A connection to the characterization of the convex hull of matrices with regular inertia is introduced. Differences between the real and the complex case are indicated.