The eigenvectors and eigenvalues of block circulant matrices had been found for real symmetric matrices with symmetric submatrices, and for block circulant matrices with circulant submatrices. The eigenvectors are now found for general block circulant matrices, including the Jordan Canonical Form for defective eigenvectors. That analysis is applied to Stephen J. Watson’s alternating circulant m...

In this paper, we propose a new low-rank tensor model based on the circulant algebra, namely, twist tensor nuclear norm or t-TNN for short. The twist tensor denotes a 3-way tensor representation to laterally store 2D data slices in order. On one hand, t-TNN convexly relaxes the tensor multi-rank of the twist tensor in the Fourier domain, which allows an efficient computation using FFT. On the o...

Boundary value methods (BVMs) for ordinary di erential equations require the solution of nonsymmetric, large and sparse linear systems. In this paper, these systems are solved by using the generalized minimal residual (GMRES) method. A block-circulant preconditioner with circulant blocks (BCCB preconditioner) is proposed to speed up the convergence rate of the GMRES method. The BCCB preconditio...

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 N blocks with M M elements per block, a block circulant matrix R is used with the same block structure as its preconditioner. In this research, the spec...

For two-dimensional (2-D) digital filters implemented by a block recursive equation, explicit relations between their frequency characteristics and those of scalar filter are obtained. Specifically, these include the relation between the discrete-time Fourier transform (DTFT) of the block recursive equation and that of the scalar 2-D difference equation, and the relation between the block matri...

A fast approximate inversion method is proposed for the block lower triangular Toeplitz with tri-diagonal blocks (BL3TB) matrix. The BL3TB matrix is approximated by a block ε-circulant matrix, which can be efficiently inverted using the fast Fourier transforms. The error estimation is given to show the high accuracy of the approximation. In applications, the proposed method is employed to solve...

We are going to study the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks. We are going also to identify the limiting spectral measure when the Hermitian block-structure is Circulant. Using the limiting spectral measure of a Hermitian Circulant block-matrix we will show that the spectral measure of a Wigner matrix with k−weakly depend...

In this paper, we investigate the reduced form of circulant matrices and we show that the problem of computing the q-th roots of a nonsingular circulant matrix A can be reduced to that of computing the q-th roots of two half size matrices B - C and B + C.