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

Abstract

Abstract. We study the asymptotic behaviour of the eigenvalues of Hermitian n × n block Toeplitz matrices An,m, with m × m Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function f , and we study their eigenvalues for large n and m, relating their behaviour to some properties of f as a function; in particular we show that, for any fixed k, the first k eigenvalues of An,m tend to inf f , while the last k tend to sup f , so extending to the block case a well-known result due to Szegö. In the case the An,m's are positive-definite, we study the asymptotic spectrum of P −1 n,m An,m, where Pn,m is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system An,mx = b, obtaining strict estimates, when n and m are fixed, and exact limit values, when n and 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.

Asymptotic eigenvalue distribution of block Toeplitz matrices and application to blind SIMO channel identification

“…Hence, its smallest nonzero eigenvalue is critical to the performance of the blind identification algorithm. We point out that in this context (blind SIMO channel identification), the herein proved result (Theorem 3) can be considered more appropriate than that of [16]- [19]. The asymptotic results proved therein are established for block Toeplitz matrices with Toeplitz blocks (BTTB) where both the size and number of blocks tend to infinity; while in this correspondence, only the size of the blocks n tends to infinity.…”

“…The number and size of blocks refer to the spatial samples of the process, possibly large. In this context, results in [16]- [19] appear better adapted.…”

Biologically Inspired Filters Utilizing Spectral Properties of Toeplitz-Block-Toeplitz Matrices

“…The relationship between the spectra of Toeplitz matrix and its generating function was introduced by Grenader and Szego [16]. Following their work, Sierra [17] and Tilli [18] showed that the interval containing the eigenvalues of the TBT matrix is closely related to the properties of the matrix generating function. More precisely, Sierra [17] states that for the function f continuous on the interval  …”

A Szegő type theorem and distribution of symplectic eigenvalues

“…where λ j (T n ), j = 1, 2, • • • , n, are the eigenvalues of T n . Many different versions and proofs of this theorem are available in the literature [2,22,24,25,26,27,28,30,31]. We prove an analogue of this theorem for symplectic eigenvalues, and apply this to compute the entropy rate and to study the distribution of symplectic eigenvalues of block Toeplitz matrices.…”