Crouzeix's conjecture
Crouzeix's conjecture is an unsolved problem in matrix analysis. It was proposed by Michel Crouzeix in 2004,[1] and it can be stated as follows:
- [math]\displaystyle{ \|f(A)\| \le 2 \sup_{z\in W(A)} |f(z)|, }[/math]
where the set [math]\displaystyle{ W(A) }[/math] is the field of values of a n×n (i.e. square) complex matrix [math]\displaystyle{ A }[/math] and [math]\displaystyle{ f }[/math] is a complex function that is analytic in the interior of [math]\displaystyle{ W(A) }[/math] and continuous up to the boundary of [math]\displaystyle{ W(A) }[/math]. Slightly reformulated, the conjecture can also be stated as follows: for all square complex matrices [math]\displaystyle{ A }[/math] and all complex polynomials [math]\displaystyle{ p }[/math]:
- [math]\displaystyle{ \|p(A)\| \le 2 \sup_{z\in W(A)} |p(z)| }[/math]
holds, where the norm on the left-hand side is the spectral operator 2-norm.
History
Crouzeix's theorem, proved in 2007, states that:[2]
- [math]\displaystyle{ \|f(A)\| \le 11.08 \sup_{z\in W(A)} |f(z)| }[/math]
(the constant [math]\displaystyle{ 11.08 }[/math] is independent of the matrix dimension, thus transferable to infinite-dimensional settings).
Michel Crouzeix and Cesar Palencia proved in 2017 that the result holds for [math]\displaystyle{ 1+\sqrt{2} }[/math],[3] improving the original constant of [math]\displaystyle{ 11.08 }[/math]. The not yet proved conjecture states that the constant can be refined to [math]\displaystyle{ 2 }[/math].
Special cases
While the general case is unknown, it is known that the conjecture holds for some special cases. For instance, it holds for all normal matrices,[4] for tridiagonal 3×3 matrices with elliptic field of values centered at an eigenvalue[5] and for general n×n matrices that are nearly Jordan blocks.[4] Furthermore, Anne Greenbaum and Michael L. Overton provided numerical support for Crouzeix's conjecture.[6]
Further reading
- Ransford, Thomas; Schwenninger, Felix L. (2018-03-01). "Remarks on the Crouzeix–Palencia Proof that the Numerical Range is a [math]\displaystyle{ (1+\sqrt2) }[/math]-Spectral Set". SIAM Journal on Matrix Analysis and Applications 39 (1): 342–345. doi:10.1137/17M1143757.
- Gorkin, Pamela; Bickel, Kelly (2018-10-27). "Numerical Range and Compressions of the Shift". arXiv:1810.11680 [math.FA].
References
- ↑ Crouzeix, Michel (2004-04-01). "Bounds for Analytical Functions of Matrices". Integral Equations and Operator Theory 48 (4): 461–477. doi:10.1007/s00020-002-1188-6. ISSN 0378-620X.
- ↑ Crouzeix, Michel (2007-03-15). "Numerical range and functional calculus in Hilbert space". Journal of Functional Analysis 244 (2): 668–690. doi:10.1016/j.jfa.2006.10.013.
- ↑ Crouzeix, Michel; Palencia, Cesar (2017-06-07). "The Numerical Range is a [math]\displaystyle{ (1+\sqrt2) }[/math]-Spectral Set". SIAM Journal on Matrix Analysis and Applications 38 (2): 649–655. doi:10.1137/17M1116672.
- ↑ 4.0 4.1 Choi, Daeshik (2013-04-15). "A proof of Crouzeix's conjecture for a class of matrices". Linear Algebra and Its Applications 438 (8): 3247–3257. doi:10.1016/j.laa.2012.12.045.
- ↑ Glader, Christer; Kurula, Mikael; Lindström, Mikael (2018-03-01). "Crouzeix's Conjecture Holds for Tridiagonal 3 x 3 Matrices with Elliptic Numerical Range Centered at an Eigenvalue". SIAM Journal on Matrix Analysis and Applications 39 (1): 346–364. doi:10.1137/17M1110663.
- ↑ Greenbaum, Anne; Overton, Michael L. (2017-05-04). "Numerical investigation of Crouzeix's conjecture". Linear Algebra and Its Applications 542: 225–245. doi:10.1016/j.laa.2017.04.035. https://cs.nyu.edu/overton/papers/pdffiles/NumerInvestCrouzeixConj.pdf.
See also
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