Diagonalization of matrices over rings

We propose a method for diagonalizing matrices with entries in commutative rings. The point of departure is to split the characteristic polynomial of the matrix over a (universal) splitting algebra, and to use the resulting universal roots to construct eigenvectors of the matrix. A crucial point is to determine when the determinant of the eigenvector matrix, that is the matrix whose columns are the eigenvectors, is regular in the splitting algebra. We show that this holds when the matrix is generic, that is, the entries are algebraically independent over the base ring. It would have been desirable to have an explicit formula for the determinant in the generic case. However, we have to settle for such a formula in a special case that is general enough for proving regularity in the general case. We illustrate the uses of our results by proving the Spectral Mapping Theorem, and by generalizing a fundamental result from classical invariant theory.