  The Cayley-Hamilton problem of expressing functions of matrices in terms of
only their eigenvalues is well-known to simplify to finding the inverse of the
confluent Vandermonde matrix. Here, we give a highly compact formula for the
inverse of any matrix, and apply it to the confluent Vandermonde matrix,
achieving in a single equation what has only been achieved by long iterative
algorithms until now. As a prime application, we use this result to get a
simple formula for explicit exponential operators in terms of only their
eigenvalues, with an emphasis on application to finite discrete quantum systems
with time dependence. This powerful result permits explicit solutions to all
Schr\"odinger and von Neumann equations for time-commuting Hamiltonians, and
explicit solutions to any degree of approximation in the non-time-commuting
case. The same methods can be extended to general finite discrete open systems
to get explicit quantum operations for time evolution using effective joint
systems, and the exact solution of all finite discrete Baker-Campbell-Hausdorff
formulas.
