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- W4300175449 abstract "We consider banded block Toeplitz matrices $T_n$ with $n$ block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of $T_n$ for $ntoinfty$ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix. For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of $T_n$ and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if $T_n$ is a Hessenberg matrix." @default.
- W4300175449 created "2022-10-03" @default.
- W4300175449 creator A5005334950 @default.
- W4300175449 date "2011-01-13" @default.
- W4300175449 modified "2023-10-18" @default.
- W4300175449 title "Equilibrium problem for the eigenvalues of banded block Toeplitz matrices" @default.
- W4300175449 doi "https://doi.org/10.48550/arxiv.1101.2644" @default.
- W4300175449 hasPublicationYear "2011" @default.
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