Geometry Of Linear Matrix Inequalities
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Geometry of Linear Matrix Inequalities
This textbook provides a thorough introduction to spectrahedra, which are the solution sets to linear matrix inequalities, emerging in convex and polynomial optimization, analysis, combinatorics, and algebraic geometry. Including a wealth of examples and exercises, this textbook guides the reader in helping to determine the convex sets that can be represented and approximated as spectrahedra and their shadows (projections). Several general results obtained in the last 15 years by a variety of different methods are presented in the book, along with the necessary background from algebra and geometry.
Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions
Author: J. William Helton
language: en
Publisher: American Mathematical Soc.
Release Date: 2019-02-21
An operator C on a Hilbert space H dilates to an operator T on a Hilbert space K if there is an isometry V:H→K such that C=V∗TV. A main result of this paper is, for a positive integer d, the simultaneous dilation, up to a sharp factor ϑ(d), expressed as a ratio of Γ functions for d even, of all d×d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space.
Matrix Inequalities and Their Extensions to Lie Groups
Matrix Inequalities and Their Extensions to Lie Groups gives a systematic and updated account of recent important extensions of classical matrix results, especially matrix inequalities, in the context of Lie groups. It is the first systematic work in the area and will appeal to linear algebraists and Lie group researchers.