The cyclotomic matrix is commonly used to arrange cyclotomic numbers in a convenient format. A natural question is whether the structure of the matrix can reflect properties of these numbers. In this article, we examine cyclotomic numbers through their associated cyclotomic matrix and reveal an algebraic structure by relating it to a basis element of a Schur ring. This viewpoint leads to structural identities and reinterpretations of classical results. As an application, we investigate the power difference set problem and establish conditions expressed through cyclotomic matrices, including spectral and determinant characterizations.