On polynomial Schur's matrix

Classical Schur's matrix is a different evaluation, provided by Schur, of the quadratic Gaussian sum from Gauss. The advanced information was studied by L. Carlitz who determined its eigenvalues, and by P. Morton who determined its eigenvectors. In this paper, we generalize the classical Schur's matrix to the case in polynomial rings over finite fields, and what is more, we give explicit formulas for the determinant, inverse matrix, eigenvalues, multiplicity and eigenvectors with respect to each eigenvalue of the polynomial Schur's matrix.