Matrices diagonalized by the discrete cosine and discrete sine transforms

We identify the matrices that are diagonalized by the discrete cosine and discrete sine transforms, respectively. Our method for the discrete cosine transform affords a simple combinatorial interpretation for the matrix entries and it yields a novel perspective. Our work on the discrete sine transform provides a generalization of $\mathscr{T}$-classes, sets of matrices of interest in the spectral theory of Toeplitz matrices. For discrete sine transform matrices, we also give a new combinatorial approach to the diagonalized matrices.