On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

Abstract

We study the asymptotic behaviour of the eigenvalues of Hermitian n × n n\times n block Toeplitz matrices A n , m A_{n,m} , with m × m m\times m Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function f f , and we study their eigenvalues for large n n and m m , relating their behaviour to some properties of f f as a function; in particular we show that, for any fixed k k , the first k k eigenvalues of A n , m A_{n,m} tend to inf f \inf f , while the last k k tend to sup f \sup f , so extending to the block case a well-known result due to Szegö. In the case the A n , m A_{n,m} ’s are positive-definite, we study the asymptotic spectrum of P n , m − 1 A n , m P_{n,m}^{-1}A_{n,m} , where P n , m P_{n,m} is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system A n , m x = b A_{n,m}x=b , obtaining strict estimates, when n n and m m are fixed, and exact limit values, when n n and m m tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.