## Abstract

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.

Original language | English |
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Pages (from-to) | 83-104 |

Number of pages | 22 |

Journal | Linear Algebra and Its Applications |

Volume | 360 |

DOIs | |

State | Published - 1 Feb 2003 |

## Keywords

- Convex invertible cones
- Convex sets of matrices with regular inertia
- Lyapunov matrix inclusion

## ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics