Basic Properties of Circulant Matrices and Anti-Circular Matrices

For simplicity, we adopt the following convention: i, j, k, n, l denote elements of N, K denotes a field, a, b, c denote elements of K, p, q denote finite sequences of elements of K, and M1, M2, M3 denote square matrices over K of dimension n. Next we state two propositions: (1) 1K · p = p. (2) (−1K) · p = −p. Let K be a set, let M be a matrix over K, and let p be a finite sequence. We say that M is line circulant about p if and only if: (Def. 1) len p = widthM and for all natural numbers i, j such that 〈i, j〉 ∈ the indices of M holds Mi,j = p(((j − i) mod len p) + 1). Let K be a set and letM be a matrix over K. We say thatM is line circulant if and only if: