Noncommutative convexity arises from linear matrix inequalities

Ando, 1979, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl., 26, 203, 10.1016/0024-3795(79)90179-4 Ball, 2005, Structured noncommutative multidimensional linear systems, SIAM J. Control Optim., 44, 1474, 10.1137/S0363012904443750 Ball, 1996, Zero–pole interpolation for meromorphic matrix functions on an algebraic curve and transfer functions of 2D systems, Acta Appl. Math., 45, 239, 10.1007/BF00047026 Ball, 1999, Zero–pole interpolation for meromorphic matrix functions on a compact Riemann surface and a matrix Fay trisecant identity, Amer. J. Math., 121, 841, 10.1353/ajm.1999.0025 Beck, 2001, On formal power series representations of uncertain systems, IEEE Trans. Automat. Control, 46, 314, 10.1109/9.905701 Berstel, 1984, Rational Series and Their Languages, 10.1007/978-3-642-73235-5 Bochnak, 1998 Bürgisser, 1997 Camino, 2003, Matrix inequalities: A symbolic procedure to determine convexity automatically, Integral Equations Operator Theory, 46, 399, 10.1007/s00020-001-1147-7 Cohn, 1971 Cook, 1979, Line bundles and homogeneous matrices, Q. J. Math., 30, 423, 10.1093/qmath/30.4.423 Cox, 1992, Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra Dixon, 1900, Note on the reduction of a ternary quartic to a symmetrical determinant, Proc. Cambridge Philos. Soc., 2, 350 Dubrovin, 1983, Matrix finite zone operators, vol. 23, 33 2000 Fliess, 1974, Matrices de Hankel, J. Math. Pure Appl., 53, 197 Fliess, 1974, Sur Divers Produits Series Formalles, Bull. Soc. Math. France, 102, 184 Gohberg, 1988, Nodes and realization of rational matrix functions: Minimality theory and applications, vol. 29, 181 Gahinet, 1995 Helton, 2002, Manipulating matrix inequalities automatically, vol. 134, 237 Helton, 2004, Convex polynomials have degree two or less, SIAM J. Matrix Anal., 25, 1124, 10.1137/S0895479803421999 Helton, 1998, Sufficient conditions for optimization of matrix functions, 1 Helton, 2006, Linear matrix inequality representation of sets, Comm. Pure Appl. Math., 59, 1 Kraus, 1936, Über konvexe Matrixfunktionen, Math. Z., 41, 18, 10.1007/BF01180403 N.M. Kumar, private communication, October 2004 Li, 2000, Extremal characterizations of the Schur complement and resulting inequalities, SIAM Rev., 42, 233, 10.1137/S0036144599337290 P.A. Linnell, Noncommutative localization in group rings, arXiv, November 2003, pp. 1–13 Löwner, 1934, Über monotone Matrixfunktionen, Math. Z., 38, 177, 10.1007/BF01170633 Nestrov, 1994, Interior Point Polynomial Algorithms in Convex Programming, vol. 13 Rowen, 1980 Schützenberger, 1961, On the definition of a family of automata, Inform. Control, 4, 245, 10.1016/S0019-9958(61)80020-X Shafarevich, 1974 Skelton, 1997 N. Slinglend, Towards computing LMIs automatically, preprint Sturm, 1999, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11–12, 625, 10.1080/10556789908805766 Valiant, 1979, Completeness classes in algebra, 249 Vinnikov, 1989, Complete description of determinantal representations of smooth irreducible curves, Linear. Algebra Appl., 125, 103, 10.1016/0024-3795(89)90035-9 Vinnikov, 1993, Self-adjoint determinantal representations of real plane curves, Math. Ann., 296, 453, 10.1007/BF01445115 Wall, 1978, Nets of quadrics, and theta-characteristics of singular curves, Philos. Trans. R. Soc. London Ser. A, 289, 229, 10.1098/rsta.1978.0060