Matrix difference equation

Summary

A matrix difference equation describes how a vector of quantities changes in discrete time by multiplication by a matrix, possibly plus an input term. It is the discrete analogue of a system of linear differential equations. Every linear recurrence of higher order can be rewritten as a first-order matrix difference equation for a state vector. With a constant matrix $A$, the homogeneous solutions are $x(n) = A^n x(0)$, so their behaviour is governed by the eigenvalues of $A$.