Theorems of the alternative for cones and Lyapunov regularity of matrices

Bryan Cain; Daniel Hershkowitz; Hans Schneider

Czechoslovak Mathematical Journal (1997)

  • Volume: 47, Issue: 3, page 487-499
  • ISSN: 0011-4642

Abstract

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Standard facts about separating linear functionals will be used to determine how two cones C and D and their duals C * and D * may overlap. When T V → W is linear and K ⊂ V and D ⊂ W are cones, these results will be applied to C = T ( K ) and D , giving a unified treatment of several theorems of the alternate which explain when C contains an interior point of D . The case when V = W is the space H of n × n Hermitian matrices, D is the n × n positive semidefinite matrices, and T ( X ) = A X + X * A yields new and known results about the existence of block diagonal X ’s satisfying the Lyapunov condition: T ( X ) is an interior point of D . For the same V , W and D , T ( X ) = X - B * X B will be studied for certain cones K of entry-wise nonnegative X ’s.

How to cite

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Cain, Bryan, Hershkowitz, Daniel, and Schneider, Hans. "Theorems of the alternative for cones and Lyapunov regularity of matrices." Czechoslovak Mathematical Journal 47.3 (1997): 487-499. <http://eudml.org/doc/30378>.

@article{Cain1997,
abstract = {Standard facts about separating linear functionals will be used to determine how two cones $C$ and $D$ and their duals $C^*$ and $D^*$ may overlap. When $T\:V\rightarrow W$ is linear and $K \subset V$ and $D\subset W$ are cones, these results will be applied to $C=T(K)$ and $D$, giving a unified treatment of several theorems of the alternate which explain when $C$ contains an interior point of $D$. The case when $V=W$ is the space $H$ of $n\times n$ Hermitian matrices, $D$ is the $n\times n$ positive semidefinite matrices, and $T(X) = AX + X^*A$ yields new and known results about the existence of block diagonal $X$’s satisfying the Lyapunov condition: $T(X)$ is an interior point of $D$. For the same $V$, $W$ and $D$, $ T(X)=X-B^*XB$ will be studied for certain cones $K$ of entry-wise nonnegative $X$’s.},
author = {Cain, Bryan, Hershkowitz, Daniel, Schneider, Hans},
journal = {Czechoslovak Mathematical Journal},
keywords = {cones; theorems of the alternative; positive semidefinite matrices; Lyapunov regularity; Lyapunov stability},
language = {eng},
number = {3},
pages = {487-499},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Theorems of the alternative for cones and Lyapunov regularity of matrices},
url = {http://eudml.org/doc/30378},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Cain, Bryan
AU - Hershkowitz, Daniel
AU - Schneider, Hans
TI - Theorems of the alternative for cones and Lyapunov regularity of matrices
JO - Czechoslovak Mathematical Journal
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 3
SP - 487
EP - 499
AB - Standard facts about separating linear functionals will be used to determine how two cones $C$ and $D$ and their duals $C^*$ and $D^*$ may overlap. When $T\:V\rightarrow W$ is linear and $K \subset V$ and $D\subset W$ are cones, these results will be applied to $C=T(K)$ and $D$, giving a unified treatment of several theorems of the alternate which explain when $C$ contains an interior point of $D$. The case when $V=W$ is the space $H$ of $n\times n$ Hermitian matrices, $D$ is the $n\times n$ positive semidefinite matrices, and $T(X) = AX + X^*A$ yields new and known results about the existence of block diagonal $X$’s satisfying the Lyapunov condition: $T(X)$ is an interior point of $D$. For the same $V$, $W$ and $D$, $ T(X)=X-B^*XB$ will be studied for certain cones $K$ of entry-wise nonnegative $X$’s.
LA - eng
KW - cones; theorems of the alternative; positive semidefinite matrices; Lyapunov regularity; Lyapunov stability
UR - http://eudml.org/doc/30378
ER -

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