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.^[1]^[2] The order of the equation is the maximum time gap between any two indicated values of the variable vector. For example,

\mathbf {x} _{t}=\mathbf {Ax} _{t-1}+\mathbf {Bx} _{t-2}

is an example of a second-order matrix difference equation, in which $x$ is an $n \times 1$ vector of variables and $A$ and $B$ are $n \times n$ 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

\mathbf {x} _{t+2}=\mathbf {Ax} _{t+1}+\mathbf {Bx} _{t}

or as

\mathbf {x} _{n}=\mathbf {Ax} _{n-1}+\mathbf {Bx} _{n-2}

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

^ Cull, Paul; Flahive, Mary; Robson, Robbie (2005). Difference Equations: From Rabbits to Chaos. Springer. ch. 7. ISBN 0-387-23234-6.
^ Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (3rd ed.). McGraw-Hill. pp. 608–612. ISBN 9780070107809.

[1] Cull, Paul; Flahive, Mary; Robson, Robbie (2005). Difference Equations: From Rabbits to Chaos. Springer. ch. 7. ISBN 0-387-23234-6.

[2] Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (3rd ed.). McGraw-Hill. pp. 608–612. ISBN 9780070107809.

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