On Wielandt-Mirsky's conjecture for matrix polynomials
Bull. Korean Math. Soc.
Published online 2019 Mar 14
Công-Trình LÊ
Quy Nhon University
Abstract : In matrix analysis, the \textit{Wielandt-Mirsky's 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 false in general. However it is true for the Frobenius distance and the Frobenius norm (the Hoffman-Wielandt's inequality). The main aim of this paper is to study the Hoffman-Wielandt's inequality and some weaker versions of the Wielandt-Mirsky's conjecture for matrix polynomials.
Keywords : Wielandt-Mirsky's conjecture, Hoffman-Wielandt's inequality, matrix polynomial, spectral variation
MSC numbers : 15A18, 15A42, 15A60, 65F15
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