  A simple closed-form solution to the discrete Lyapunov equation (DLE) is
established for certain families of matrices. This solution is expressed in
terms of the eigen decomposition (ED) for which closed-form solutions are known
for all 2x2; 3x3 and certain families of matrices. For general matrices, the
proposed ED based closed-form solution can be used as an efficient numerical
solution when the ED can be computed. The result is then extended to give
closed-form solutions for a generalization of the DLE, called the Stein
equation. The proposed explicit solution's complexity is of the same order as
iterative solutions and significantly smaller than known closed-form solutions.
These solutions may prove convenient for analysis and synthesis problems
related to these equations due to their compact form.
