NONCOMMUTATIVE CONVEXITY ARISES FROM LINEAR MATRIX INEQUALITIES.

By J. William Helton, Scott McCullough, Victor Vinnikov. 2006

136 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/ca17457b5b2a2a805e6e84b8f6821342d3d73e00

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In Applied Mathematics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Singularities of rational functions and minimal factorizations: The noncommutative and the commutative setting (2009, background): “State space realizations are exactly what is needed to convert (numerically unmanageable) rational matrix inequalities into (highly manageable) linear matrix inequalities (see [31]).”
Polynomials with and without determinantal representations (2010, methodology): “Helton and Vinnikov conjectured however that their result is true if one allows for matrices of size larger than d. Brändén [2] has now recently disproved this conjecture.”
Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method (2015, background): “(Versions of this l nearization trick have a long history in different settings, see [7, 19, 18].”
LMI Representations of Convex Semialgebraic Sets and Determinantal Representations of Algebraic Hypersurfaces: Past, Present, and Future (2012, unknown): “4 was first established in [27] using free noncommutative techniques.”
Foundations of Free Noncommutative Function Theory (2012, background): “, [74, 73, 98] — in the context of noncommutative convexity and noncommutative real semialgebraic geometry, where one considers polynomials and rational functions in d noncommuting indeterminates evaluated on d-tuples of matrices over R [49, 56, 59, 58, 54, 57].”
Operator Theory on Noncommutative Domains (2007, unknown): “In a related area of research, we remark the work of Helton, McCullough, Putinar, and Vinnikov, on symmetric noncommutative polynomials ([48], [49], [51], [50], [52]), which contains a new type of engineering application.”
Noncommutative reproducing kernel Hilbert spaces (2016, background): “ependently in the work of Voiculescu and collaborators (see [49, 50]) in the context of the needs of a free-probability theory, as well as in the work of Helton-Klep-McCullough and collaborators (see [25, 26, 27, 29, 11]) on free noncommutative Linear-Matrix-Inequality relaxations and connections wi”
Free holomorphic functions and interpolation (2007, methodology): “In related areas of research, we remark the work of Helton, McCullough, Putinar, and Vinnikov, on symmetric noncommutative polynomials ([13], [14], [15], [16]), and the work of Muhly and Solel on representations of tensor algebras over C correspondences (see [19], [20]).”