An n × n matrix D = d[i, j] is said to be circulant, if the entries d[i, j] verifying (j − i) = k mod n, for some k, have the same value (for a survey on circulant matrix properties, see Davis (1979)). A directed (respectively, undirected) graph is circulant, if its adjacency matrix is circulant (respectively, symmetric, and circulant). Similarly, a weighted graph is circulant, if its weighted ...

We propose geometrical methods for constructing square 01-matrices with the same number n of units in every row and column, and such that any two rows of the matrix have at most one unit in the same position. In terms of Design Theory, such a matrix is an incidence matrix of a symmetric configuration. Also, it gives rise to an n-regular bipartite graphs without 4-cycles, which can be used for c...

We associate to any given circulant complex matrix C another one E(C) such that E(E(C)) = C∗ the transpose conjugate of C. All circulant Hadamard matrices of order 4 satisfy a condition C4 on their eigenvalues, namely, the absolute value of the sum of all eigenvalues is bounded above by 4. We prove by a “descent” that uses our operator E that the only circulant Hadamard matrices of order n > 4,...

Optimal preconditioners (those that provide a proper cluster at 1) are very important for the cg-like methods since they make them converge superlinearly. As is well known, for Toeplitz matrices generated by a continuous symbol, many circulant and circulant-like (related to different matrix algebras) preconditioners were proved to be optimal. In contrast, for multilevel Toeplitz matrices, there...

We investigate two-sided bounds for operator norms of random matrices with non-homogenous independent entries. formulate a lower bound Rademacher and conjecture that it may be reversed up to universal constant. show our holds loglogn factor randomized n×n circulant the double logarithm eliminated under some mild additional assumptions on coefficients.

A new method of obtaining a sequence isolated complex Hadamard matrices (CHM) for dimensions $N\geqslant 7$, based on block-circulant structures, is presented. We discuss, several analytic examples resulting from modification the Sinkhorn algorithm. In particular, we present orders $9$, $10$ and $11$, which elements are not roots unity, also multiparametric families order $10$. note novel conne...

Circulant preconditioners that cluster eigenvalues are well established for linear systems involving symmetric positive definite Toeplitz matrices. For these preconditioners rapid convergence of the preconditioned conjugate gradient method is guaranteed. Since circulant preconditioners can be applied quickly using the fast Fourier transform, preconditioned CG with circulant preconditioning is e...

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...