Geodesic
Theorems of the alternative for cones and Lyapunov regularity of matrices
Czechoslovak Mathematical Journal, Tome 47 (1997) no. 3, pp. 487-499 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $T\:V\rightarrow W$ 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.
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.
Classification : 15A24, 15A48, 15A57, 46A40, 46N10, 52A05, 90C48 
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Cain, Bryan; Hershkowitz, Daniel; Schneider, Hans. Theorems of the alternative for cones and Lyapunov regularity of matrices. Czechoslovak Mathematical Journal, Tome 47 (1997) no. 3, pp. 487-499. http://geodesic.mathdoc.fr/item/CMJ_1997_47_3_a8/

[1] G. P. Barker, A. Berman, and R. J. Plemmons: Positive diagonal solutions to the Lyapunov equations. Lin. Multilin. Alg. 5 (1978), 249–256. | DOI | MR

[2] G. P. Barker, B. S. Tam, and Norbil Davila: A geometric Gordan-Stiemke theorem. Lin. Alg. Appl. 61 (1984), 83–89. | DOI | MR

[3] A. Ben-Israel: Linear equations and inequalities on finite dimensional, real or complex, vector spaces: A unified theory. Math. Anal. Appl. 27 (1969), 367–389. | DOI | MR | Zbl

[4] A. Berman: Cones, Matrices and Mathematical Programming. Lecture Notes in Economics and Mathematical Systems, Vol. 79, Springer-Verlag, 1973. | MR | Zbl

[5] A. Berman and A. Ben-Israel: More on linear inequalities with application to matrix theory. J. Math. Anal. Appl. 33 (1971), 482–496. | DOI | MR

[6] A. Berman and R. C. Ward: ALPS: Classes of stable and semipositive matrices. Lin. Alg. Appl. 21 (1978), 163–174. | DOI | MR

[7] D. H. Carlson, D. Hershkowitz, and D. Shasha: Block diagonal semistability factors and Lyapunov semistability of block triangular matrices. Lin. Alg. Appl. 172 (1992), 1–25. | DOI | MR

[8] D. H. Carlson and H. Schneider: Inertia theorems for matrices: the semidefinite case. J. Math. Anal. Appl. 6 (1963), 430–436. | DOI | MR

[9] A. Ja. Dubovickii and A.A. Miljutin: Extremum problems with certain constraints. Soviet Math. 4 (1963), 759–762. | MR

[10] D. Gale: The theory of linear economic models. McGraw-Hill, 1960. | MR

[11] I.V. Girsanov: Lectures on Mathematical Theory of Extremum Problems [sic]. Lecture Notes in Economics and Mathematical Systems, Vol. 67, Springer-Verlag, 1972. | MR

[12] D. Hershkowitz and H. Schneider: Semistability factors and semifactors. Contemp. Math. 47 (1985), 203–216. | DOI | MR

[13] J. L. Kelley, I. Namioka, et al.: Linear Topological Spaces. van Nostrand, 1963. | MR

[14] A. N. Lyapunov: Le problème général de la stabilité du mouvement. Ann. Math. Studies 17 (1949), Princeton University Press.

[15] H. Nikaido: Convex Structures and Economic Theory. Mathematics in Science and Engineering, Vol. 51, Academic, 1968. | MR | Zbl

[16] A. Ostrowski and H. Schneider: Some theorems on the inertia of general matrices. J. Math. Analysis and Appl. 4 (1962), 72–84. | DOI | MR

[17] R. T. Rockafellar: Convex Analysis. Princeton University Press, 1970. | MR | Zbl

[18] B. D. Saunders and H. Schneider: Applications of the Gordan-Stiemke theorem in combinatorial matrix theory. SIAM Rev. 21 (1979), 528–541. | DOI | MR

[19] O. Taussky: Matrices $C$ with $C^n \rightarrow 0$. J. Alg. 1 (1964), 1–10. | MR | Zbl

[20] E. Zeidler: Nonlinear functional analysis and its applications III: Variational methods and optimation. Springer-Verlag, 1985. | MR