A direct solution to the linear variance equation of a time-invariant linear system

A closed form solution of the linear variance equation, which is amenable to machine computation, is presented. It allows one to compute the covariance matrix associated with the linear system starting at any initial condition without using recursive techniques. Thus a large amount of computing time can be saved if one is interested in the covariance matrix at only a few times. Also, one can obtain the steady-state result directly. The primary restriction is that the system of interest must have a system matrix which is diagonalizable.