On Wielandt-Mirsky's conjecture for matrix polynomials
Bull. Korean Math. Soc. 2019 Vol. 56, No. 5, 1273-1283
https://doi.org/10.4134/BKMS.b181065
Published online September 30, 2019
C\^ong-Tr\`inh L\^e
Ton Duc Thang University
Abstract : In matrix analysis, the \textit{Wielandt-Mirsky conjecture} states that $$ \dist(\sigma(A), \sigma(B)) \leq \|A-B\| $$ for any normal matrices $ A, B \in \mathbb C^{n\times n}$ and any operator norm $\|\cdot \|$ on $C^{n\times n}$. Here $\dist(\sigma(A), \sigma(B))$ denotes the optimal matching distance between the spectra of the matrices $A$ and $ B$. It was proved by A.~J.~Holbrook (1992) that this conjecture is \textit{false } in general. However it is true for the Frobenius distance and the Frobenius norm (the \textit{Hoffman-Wielandt inequality}). The main aim of this paper is to study the Hoffman-Wielandt inequality and some weaker versions of the Wielandt-Mirsky conjecture for matrix polynomials.
Keywords : Wielandt-Mirsky's conjecture, Hoffman-Wielandt's inequality, Wielandt's inequality, matrix polynomial, spectral variation
MSC numbers : Primary 15A18, 15A42, 15A60, 65F15
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