| 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. (en)
|
| dbo:wikiPageID
| |
| dbo:wikiPageLength
|
- 4289 (xsd:nonNegativeInteger)
|
| dbo:wikiPageRevisionID
| |
| dbo:wikiPageWikiLink
| |
| dbp:wikiPageUsesTemplate
| |
| dct:subject
| |
| 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 : (en)
|
| rdfs:label
|
- Crouzeix's conjecture (en)
|
| owl:sameAs
| |
| prov:wasDerivedFrom
| |
| foaf:isPrimaryTopicOf
| |
| is dbo:wikiPageRedirects
of | |
| is dbo:wikiPageWikiLink
of | |
| is foaf:primaryTopic
of | |
