On the resolvent condition in the Kreiss Matrix Theorem | Zendy

Randall J. LeVeque | Zendy; Lloyd N. Trefethen | Zendy

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On the resolvent condition in the Kreiss Matrix Theorem

Author(s) -

Randall J. LeVeque,

Lloyd N. Trefethen

Publication year - 1984

Publication title -

bit numerical mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.904

H-Index - 59

eISSN - 1572-9125

pISSN - 0006-3835

DOI - 10.1007/bf01934916

Subject(s) - resolvent , mathematics , matrix (chemical analysis) , lemma (botany) , complex plane , matrix function , mathematical analysis , mathematical proof , pure mathematics , combinatorics , symmetric matrix , eigenvalues and eigenvectors , geometry , physics , materials science , composite material , ecology , poaceae , quantum mechanics , biology

The Kreiss Matrix Theorem asserts the uniform equivalence over allN ×N matrices of power boundedness and a certain resolvent estimate. We show that the ratio of the constants in these two conditions grows linearly withN, and we obtain the optimal proportionality factor up to a factor of 2. Analogous results are also given for the related problem involving matrix exponentialseAt. The proofs make use of a lemma that may be of independent interest, which bounds the arc length of the image of a circle in the complex plane under a rational function.

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