In this paper, we propose a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices An. The n 2 th column of our circulant preconditioner Sn is equal to the n 2 th column of the given matrix An. Thus if An is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be deened as (S n Sn)...

Calculating the permanent of a (0, 1) matrix is a #P -complete problem but there are some classes of structuredmatrices for which the permanent is calculable in polynomial time. The most well-known example is the fixedjump (0, 1) circulant matrix which, using algebraic techniques, was shown by Minc to satisfy a constant-coefficient fixed-order recurrence relation. In this note we show how, by i...

We consider circulant preconditioners for Hermitian Toeplitz systems from the view point of function theory. We show that some well-known circulant preconditioners can be derived from convoluting the generating function f of the Toeplitz matrix with famous kernels like the Dirichlet and the Fej er kernels. Several circulant precondition-ers are then constructed using this approach. Finally, we ...

A k x k matrix A = [aU lover a field F is called circulant if aij = a (j-i) mod k' A [2k ,k l linear code over F = GF (q) is called double-circulant if it is generated by a matrix of the fonn [I A l, where A is a circulant matrix. In this work we ftrst employ the Fourier transform techJ nique to analyze and construct se:veral families of double-circulant codes. The minimum distance of the resul...

We discuss a generalization of the Cohn–Umans method, a potent technique developed for studying the bilinear complexity of matrix multiplication by embedding matrices into an appropriate group algebra. We investigate how the Cohn–Umans method may be used for bilinear operations other than matrix multiplication, with algebras other than group algebras, and we relate it to Strassen’s tensor rank ...

This work is devoted to a systematic investigation of triangular matrix forms of the Pascal Triangle. To start, the twelve matrix forms (collectively referred to as G-matrices) are presented. To highlight one way in which the G-matrices relate to each other, a set of four operators named circulant operators is introduced. These operators provide a new insight into the structure of the space of ...

A generic matrix A ∈ Cn×n is shown to be the product of circulant and diagonal matrices with the number of factors being 2n−1 at most. The demonstration is constructive, relying on first factoring matrix subspaces equivalent to polynomials in a permutation matrix over diagonal matrices into linear factors. For the linear factors, the sum of two scaled permutations is factored into the product o...

Spectral properties of sign symmetric matrices are studied. A criterion for sign symmetry of shifted basic circulant permutation matrices is proven, and is then used to answer the question which complex numbers can serve as eigenvalues of sign symmetric 3 × 3 matrices. The results are applied in the discussion of the eigenvalues of QM -matrices. In particular, it is shown that for every positiv...

An r × r matrix A = [aij] over a field F is called circulant if aij = a0, ( j−i) mod r . An [n = 2r, k = r] linear code over F = GF(q) is called double-circulant if it is generated by a matrix of the form [I A], where A is an r × r circulant matrix. In this work we first employ the Fourier transform technique to analyze and construct several families of double-circulant codes. The minimum dista...

Anti-circulant tensors have applications in exponential data fitting. They are special Hankel tensors. In this paper, we extend the definition of anti-circulant tensors to generalized anticirculant tensors by introducing a circulant index r such that the entries of the generating vector of a Hankel tensor are circulant with module r. In the special case when r=n, where n is the dimension of the...