An \(n\times n\) unitary matrix \(P\) such that \(P^ k=I\) for some integer \(k>0\) is here called a circulation matrix. An \(n\times n\) matrix \(W\) such that \(W=e^{i\theta} P^* WP\) is said to be circulative of degree \(\theta\) with respect to the circulation matrix \(P\). The author's main result is a decomposition of a matrix circulative with respect to a power \(P^ m\) into a sum of \(m\) matrices circulative with respect to \(P\). Three applications to physical problems are explained in some detail.