The circulant operator in the Banach algebra of matrices (Q753905)

Given an \(n\times n\) matrix \(A_ n\), it is observed that the circulant matrix \(c(A_ n)\) that minimizes \(\| A_ n-C_ n\|_ F\) over all \(n\times n\) circulant matrices \(C_ n\) is given by \(c(A_ n)=\sum^{n-1}_{j=0}(\frac{1}{n}\sum_{p-q\equiv (mod n)}a_{pq})Q^ j\) where Q is the full cycle permutation matrix with \(Q_{ij}=1\) if i-j\(\equiv 1(mod n)\). Here \(\| \cdot \|_ F\) is the Frobenius norm. It is shown that if both \(A_ n\) and \(c(A_ n)\) are nonsingular then \(\| I_ n-C_ n^{-1}A_ n\|_ F\) is a minimum over all \(n\times n\) nonsingular circulant matrices \(C_ n\) for \(C_ n=c(A_ nA^*_ n)\subset (A^*_ n)^{-1}\). Here \(I_ n\) is the \(n\times n\) identity matrix. Several spectral properties of \(c(A_ n)\) are discussed.