  We present a prescription for forming matrices with specified eigenvalues and
known eigenvectors. With this method, we can form Hermitian, anti-Hermitian,
symmetric and general matrices with arbitrary eigenvalues. In addition we
propose an algorithm for diagonalizing such matrices. The functions required
for the realization of this are probability amplitudes connecting observables
with discrete eigenvalue spectra and can be obtained from spin theory. For the
example case of $5\times 5$ matrices, these functions are given.
