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October 2019 On uniform connectivity of algebraic matrix sets
Fredy Vides
Banach J. Math. Anal. 13(4): 918-943 (October 2019). DOI: 10.1215/17358787-2019-0009


We study the uniform path connectivity of sets of matrix tuples that satisfy some additional constraints, and more specifically, given ε>0, a fixed metric ð in Mn(C)m induced by the operator norm , any collection of r nonconstant polynomials p1(x1,,xm),,pr(x1,,xm) over C with finite zero set Z(p1,,pr)Cm and any m-tuple X=(X1,,Xm) in the set ZDnm(p1,,pr)Mnm(C) of commuting normal matrix contractions such that pj(Y1,,Ym)=0 for each (Y1,,Ym)ZDnm(p1,,pr) and each 1jr. The author proves the existence of paths between arbitrary m-tuples that belong to the intersection of ZDnm(p1,,pr) and the open δ-ball Bð(X,δ) centered at X for some δ>0 that can be chosen independently of n. In addition, the author proves that the aforementioned paths are contained in the intersection of Bð(X,ε) and ZDnm(p1,,pr). Some connections of the main results with structure-preserving perturbation theory and preconditioning techniques are outlined.


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Fredy Vides. "On uniform connectivity of algebraic matrix sets." Banach J. Math. Anal. 13 (4) 918 - 943, October 2019.


Received: 24 June 2018; Accepted: 12 February 2019; Published: October 2019
First available in Project Euclid: 9 October 2019

zbMATH: 07118768
MathSciNet: MR4016903
Digital Object Identifier: 10.1215/17358787-2019-0009

Primary: 47N40
Secondary: 15A27 , 15A60 , 47A58

Keywords: eigenvalue clustering , functional calculus , joint spectrum , matrix function , matrix path

Rights: Copyright © 2019 Tusi Mathematical Research Group


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Vol.13 • No. 4 • October 2019
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