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Subject Item
dbr:Crouzeix's_conjecture
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Crouzeix's conjecture
rdfs:comment
Crouzeix's conjecture is an unsolved (as of 2018) problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it refines Crouzeix's theorem, which states: where the set is the field of values of a n×n (i.e. square) complex matrix and is a complex function, that is analytic in the interior of and continuous up to the boundary of . The constant is independent of the matrix dimension, thus transferable to infinite-dimensional settings. The not yet proved conjecture states that the constant is sharpable to :
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dbc:Unsolved_problems_in_mathematics dbc:Conjectures dbc:Matrix_theory
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1068641948
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dbo:abstract
Crouzeix's conjecture is an unsolved (as of 2018) problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it refines Crouzeix's theorem, which states: where the set is the field of values of a n×n (i.e. square) complex matrix and is a complex function, that is analytic in the interior of and continuous up to the boundary of . The constant is independent of the matrix dimension, thus transferable to infinite-dimensional settings. The not yet proved conjecture states that the constant is sharpable to : Michel Crouzeix and Cesar Palencia proved in 2017 that the result holds for , improving the original constant of . Slightly reformulated, the conjecture can be stated as follows: For all square complex matrices and all complex polynomials : holds, where the norm on the left-hand side is the spectral operator 2-norm. While the general case is unknown, it is known that the conjecture holds for tridiagonal 3×3 matrices with elliptic field of values centered at an eigenvalue and for general n×n matrices that are nearly Jordan blocks. Furthermore, Anne Greenbaum and Michael L. Overton provided numerical support for Crouzeix's conjecture.
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