Localization of the Eigenvalues of Toeplitz
نویسنده
چکیده
This paper explores the relationship between Toeplitz and circulant matrices. Upper and lower bounds for all eigenvalues of hermitian Toeplitz matrices are given, capitalizing on the possibility of embedding a Toeplitz matrix in a circulant, and of expressing any nn Toeplitz matrix as a sum of two matrices with known eigenvalues. The bounds can be simultaneously found using a single discrete Fourier transform evaluation. Thus, the total computation time is O(log 2 n) per bound. Simulation results indicate that the bounds are sharper than a few other known bounds. 1. INTRODUCTION The class of Toeplitz matrices is extremely important , for a number of theoretical and practical reasons. These matrices arise naturally in a variety of problems, including trigonometric moment problems, optimum ltering, linear prediction and spectral estimation.
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