Matrix difference equation

A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. The order of the equation is the maximum time gap between any two indicated values of the variable vector. For example,

is an example of a second-order matrix difference equation, in which x is an n × 1 vector of variables and A and B are n×n matrices. This equation is homogeneous because there is no vector constant term added to the end of the equation. The same equation might also be written as

or as

The most commonly encountered matrix difference equations are first-order.

An example of a non-homogeneous first-order matrix difference equation is

with additive constant vector b. The steady state of this system is a value x* of the vector x which, if reached, would not be deviated from subsequently. x* is found by setting ${\displaystyle x_{t}=x_{t-1}=x^{*}}$ in the difference equation and solving for x* to obtain

where ${\displaystyle I}$ is the n×n identity matrix, and where it is assumed that ${\displaystyle [I-A]}$ is invertible. Then the non-homogeneous equation can be rewritten in homogeneous form in terms of deviations from the steady state:

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