On the Spectrum of a Family of Preconditioned Block Toeplitz Matrices

Abstract

Research on preconditioning Toeplitz matrices with circulant matrices has been active recently. The preconditioning technique can be easily generalized to block Toeplitz matrices. That is, for a block Toeplitz matrix T consisting of $N \times N$ blocks with $M \times M$ elements per block, a block circulant matrix R is used with the same block structure as its preconditioner. In this research, the spectral clustering property of the preconditioned matrix $R^{ - 1} T$ with T generated by two-dimensional rational functions $T(z_x ,z_y )$ of order $(p_x ,q_x ,p_y ,q_y )$ is examined. It is shown that the eigenvalues of $R^{ - 1} T$ are clustered around unity except at most $O(M\gamma _y + N\gamma _x )$ outliers, where $\gamma _x = \max (p_x ,q_x )$ and $\gamma _y = \max (p_y ,q_y )$. Furthermore, if T is separable, the outliers are clustered together such that $R^{ - 1} T$ has at most $(2\gamma _x + 1)(2\gamma _y + 1)$ asymptotic distinct eigenvalues. The superior convergence behavior of the preconditioned c...