[1][2] The order of the equation is the maximum time gap between any two indicated values of the variable vector.
An example of a nonhomogeneous first-order matrix difference equation is with additive constant vector b.
x* is found by setting xt = xt−1 = x* in the difference equation and solving for x* to obtain where I is the n × n identity matrix, and where it is assumed that [I − A] is invertible.
Then the nonhomogeneous equation can be rewritten in homogeneous form in terms of deviations from the steady state: The first-order matrix difference equation [xt − x*] = A[xt−1 − x*] is stable—that is, xt converges asymptotically to the steady state x*—if and only if all eigenvalues of the transition matrix A (whether real or complex) have an absolute value which is less than 1.
Then we can iterate and substitute repeatedly from the initial condition y0, which is the initial value of the vector y and which must be known in order to find the solution: and so forth, so that by mathematical induction the solution in terms of t is Further, if A is diagonalizable, we can rewrite A in terms of its eigenvalues and eigenvectors, giving the solution as where P is an n × n matrix whose columns are the eigenvectors of A (assuming the eigenvalues are all distinct) and D is an n × n diagonal matrix whose diagonal elements are the eigenvalues of A.
This solution motivates the above stability result: At shrinks to the zero matrix over time if and only if the eigenvalues of A are all less than unity in absolute value.
Starting from the n-dimensional system yt = Ayt−1, we can extract the dynamics of one of the state variables, say y1.
Then denoting the 2n × 1 stacked vector of current and once-lagged variables as zt and the 2n × 2n block matrix as L, we have as before the solution Also as before, this stacked equation, and thus the original second-order equation, are stable if and only if all eigenvalues of the matrix L are smaller than unity in absolute value.
[4] In most contexts the evolution of H backwards through time is stable, meaning that H converges to a particular fixed matrix H* which may be irrational even if all the other matrices are rational.