Linear Matrix Pencils and Noncommutative Convexity

Jurij Volčič ()
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Jurij Volčič: University of Auckland, Department of Mathematics

Chapter 88 in Operator Theory, 2026, pp 2759-2772 from Springer

Abstract: Abstract Hermitian linear matrix pencils are ubiquitous in control theory, operator systems, semidefinite optimization, and real algebraic geometry. This survey reviews the fundamental features of the matricial solution set of a linear matrix inequality, the free spectrahedron, from the perspective of free real algebraic geometry. Namely, among matricial solution sets of noncommutative polynomial inequalities, free spectrahedra are precisely the convex ones. Furthermore, a procedure for detecting free spectrahedra and producing their representing linear matrix pencils is discussed. Finally, free spectrahedra admit a perfect Positivstellensatz, leading to a semidefinite programming formulation of eigenvalue optimization over convex matricial sets constrained by noncommutative polynomial inequalities.

Keywords: Linear matrix inequality; Matrix convexity; Linear pencil; Free spectrahedron; Free semialgebraic set; Positivstellensatz; 47A63; 14P10; 15A22; 13J30; 52A05 (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16356-1_85

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DOI: 10.1007/978-3-032-16356-1_85

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