On the extreme eigenvalues of hermitian (block) toeplitz matrices

✍ Scribed by Stefano Serra

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 758 KB
• Volume: 270
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(97)00231-0

✦ Synopsis

We are concerned with the behavior of the minimum (maximum) eigenvalue A~0 "~ (A~ "~) of an (n + 1) X (n + 1) Hermitian Toeplitz matrix T~(f) where f is an integrable real-valued function. Kac, Murdoch, and Szeg5, Widom, Patter, and R. H.

Chan obtained that A}~ 0 -rain f = O(1/n 2k) in the case where f ~ C zk, at least locally, and f -inff has a zero of order 2k. We obtain the same result under the second hypothesis alone. Moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices, proving that the rate of convergence of A~ ~ to inff depends only on the order p (not necessarily even or integer or finite) of the zero of f -inf f. With the help of this tool, we derive an absolute lower bound for the minimal eigenvalues of Toeplitz matrices generated by nonnegative L 1 functions and also an upper bound for the associated Euclidean condition numbers. Finally, these results are extended to the case of Hermitian block Toeplitz matrices with Toeplitz blocks generated by a bivariate integrable function f.