The circulant operator in the banach algebra of matrices

We study an operator c which maps every n-by-n matrix A_n to a circulant matrix c(A_n) that minimizes the Frobenius norm ‖A_n – C_n‖_F over all n-by-n circulant matrices C_n. The circulant matrix c(A_n), called the optimal circulant preconditioner, has proved to be a good preconditioner for a general class of Toeplitz systems. In this paper, we give different formulations of the operator, discuss its algebraic and geometric properties, and compute its operator norms in different Banach algebras of matrices. Using these results, we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of ‖I – C_n^-1A_n‖_F over all nonsingular circulant matrices C_n.