A note on preconditioned block Toeplitz matrices

In [SIAM J. Sci. Statist. Comput., 13 (1992), pp. 948-966], Ku and Kuo proposed and analysed a block circulant preconditioner R(mn) for solving a family of block Toeplitz systems T-mn v = b. For a special class of block matrices called the quadrantally symmetric Toeplitz matrices, they proved that the eigenvalues of R(mn)(-1)T(mn) are clustered around one except at most O(m + n) outliers with T-mn generated by a two-dimensional rational function. The superior convergence rate of the preconditioned conjugate gradient (PCG) method is explained by the clustering property of the spectrum of R(mn)(-1)T(mn). However, in their analysis, there is no discussion on the positive definiteness of the matrix T-mn, and the preconditioner R(mn), is assumed to be invertible. In this paper, we give some results on these two aspects. Under the assumptions in the above-referenced paper, we prove that if the generating function f(x, y) of T-mn is positive, then T-mn is positive definite. Moreover, we show that R(mn) is uniformly invertible when m and n are sufficiently large.