TY - JOUR

T1 - A pair of matrices sharing common Lyapunov solutions - A closer look

AU - Cohen, Nir

AU - Lewkowicz, Izchak

N1 - Funding Information:
Funded by CNPq grant no. 300019/96-3. ∗Corresponding author. Tel.: +972-8-647-2406; fax: +972-8-647-2949. E-mail addresses: nir@ime.unicamp.br (N. Cohen), izchak@ee.bgu.ac.il (I. Lewkowicz).

PY - 2003/2/1

Y1 - 2003/2/1

N2 - 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.

AB - 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.

KW - Convex invertible cones

KW - Convex sets of matrices with regular inertia

KW - Lyapunov matrix inclusion

UR - http://www.scopus.com/inward/record.url?scp=84867978054&partnerID=8YFLogxK

U2 - 10.1016/S0024-3795(02)00443-3

DO - 10.1016/S0024-3795(02)00443-3

M3 - Article

AN - SCOPUS:84867978054

VL - 360

SP - 83

EP - 104

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -