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 → 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) = 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.
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References
G. P. Barker, A. Berman, and R. J. Plemmons: Positive diagonal solutions to the Lyapunov equations. Lin. Multilin. Alg. 5 (1978), 249–256.
G. P. Barker, B. S. Tam, and Norbil Davila: A geometric Gordan-Stiemke theorem. Lin. Alg. Appl. 61 (1984), 83–89.
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.
A. Berman: Cones, Matrices and Mathematical Programming. Lecture Notes in Economics and Mathematical Systems, Vol. 79. Springer-Verlag, 1973.
A. Berman and A. Ben-Israel: More on linear inequalities with application to matrix theory. J. Math. Anal. Appl. 33 (1971), 482–496.
A. Berman and R. C. Ward: ALPS: Classes of stable and semipositive matrices. Lin. Alg. Appl. 21 (1978), 163–174.
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.
D. H. Carlson and H. Schneider: Inertia theorems for matrices: the semidefinite case. J. Math. Anal. Appl. 6 (1963), 430–436.
A. Ja. Dubovickii and A.A. Miljutin: Extremum problems with certain constraints. Soviet Math. 4 (1963), 759–762; Dokl. Akad. Nauk SSSR 149 (1963), 759–762.
D. Gale: The theory of linear economic models. McGraw-Hill, 1960.
I.V. Girsanov: Lectures on Mathematical Theory of Extremum Problems [sic]. Lecture Notes in Economics and Mathematical Systems, Vol. 67. Springer-Verlag, 1972.
D. Hershkowitz and H. Schneider: Semistability factors and semifactors. Contemp. Math. 47 (1985), 203–216.
J. L. Kelley, I. Namioka, et al.: Linear Topological Spaces. van Nostrand, 1963.
A. N. Lyapunov: Le problème général de la stabilité du mouvement. Ann. Math. Studies 17 (1949). Princeton University Press.
H. Nikaido: Convex Structures and Economic Theory. Mathematics in Science and Engineering, Vol. 51. Academic, 1968.
A. Ostrowski and H. Schneider: Some theorems on the inertia of general matrices. J. Math. Analysis and Appl. 4 (1962), 72–84.
R. T. Rockafellar: Convex Analysis. Princeton University Press, 1970.
B. D. Saunders and H. Schneider: Applications of the Gordan-Stiemke theorem in combinatorial matrix theory. SIAM Rev. 21 (1979), 528–541.
O. Taussky: Matrices C with C n → 0. J. Alg. 1 (1964), 1–10.
E. Zeidler: Nonlinear functional analysis and its applications III: Variational methods and optimation. Springer-Verlag, 1985.
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Cain, B., Hershkowitz, D. & Schneider, H. Theorems of the Alternative for Cones and Lyapunov Regularity of Matrices. Czechoslovak Mathematical Journal 47, 487–499 (1997). https://doi.org/10.1023/A:1022463518098
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DOI: https://doi.org/10.1023/A:1022463518098