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.

Nonhomogeneous first-order case and the steady state

An example of a nonhomogeneous first-order matrix difference equation is

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

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 $x t = x t -1 = x *$ in the difference equation and solving for $x *$ to obtain

\mathbf {x} ^{*}=[\mathbf {I} -\mathbf {A} ]^{-1}\mathbf {b}

where $I$ is the $n \times n$ 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:

\left[\mathbf {x} _{t}-\mathbf {x} ^{*}\right]=\mathbf {A} \left[\mathbf {x} _{t-1}-\mathbf {x} ^{*}\right]

Stability of the first-order case

The first-order matrix difference equation $[x t - x *] = A [x t -1 - x *]$ is stable—that is, $x t$ 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.

Solution of the first-order case

Assume that the equation has been put in the homogeneous form $y t = Ay t -1$ . Then we can iterate and substitute repeatedly from the initial condition $y 0$ , which is the initial value of the vector $y$ and which must be known in order to find the solution:

{\begin{aligned}\mathbf {y} _{1}&=\mathbf {Ay} _{0}\\\mathbf {y} _{2}&=\mathbf {Ay} _{1}=\mathbf {A} ^{2}\mathbf {y} _{0}\\\mathbf {y} _{3}&=\mathbf {Ay} _{2}=\mathbf {A} ^{3}\mathbf {y} _{0}\end{aligned}}

and so forth, so that by mathematical induction the solution in terms of $t$ is

\mathbf {y} _{t}=\mathbf {A} ^{t}\mathbf {y} _{0}

Further, if $A$ is diagonalizable, we can rewrite $A$ in terms of its eigenvalues and eigenvectors, giving the solution as

\mathbf {y} _{t}=\mathbf {PD} ^{t}\mathbf {P} ^{-1}\mathbf {y} _{0},

where $P$ is an $n \times n$ matrix whose columns are the eigenvectors of $A$ (assuming the eigenvalues are all distinct) and $D$ is an $n \times n$ diagonal matrix whose diagonal elements are the eigenvalues of $A$ . This solution motivates the above stability result: $A t$ shrinks to the zero matrix over time if and only if the eigenvalues of $A$ are all less than unity in absolute value.

Extracting the dynamics of a single scalar variable from a first-order matrix system

Starting from the $n$ -dimensional system $y t = Ay t -1$ , we can extract the dynamics of one of the state variables, say $y 1$ . The above solution equation for $y t$ shows that the solution for $y 1, t$ is in terms of the $n$ eigenvalues of $A$ . Therefore the equation describing the evolution of $y 1$ by itself must have a solution involving those same eigenvalues. This description intuitively motivates the equation of evolution of $y 1$ , which is

y_{1,t}=a_{1}y_{1,t-1}+a_{2}y_{1,t-2}+\dots +a_{n}y_{1,t-n}

where the parameters $a i$ are from the characteristic equation of the matrix $A$ :

\lambda ^{n}-a_{1}\lambda ^{n-1}-a_{2}\lambda ^{n-2}-\dots -a_{n}\lambda ^{0}=0.

Thus each individual scalar variable of an $n$ -dimensional first-order linear system evolves according to a univariate $n$ th-degree difference equation, which has the same stability property (stable or unstable) as does the matrix difference equation.

Solution and stability of higher-order cases

Matrix difference equations of higher order—that is, with a time lag longer than one period—can be solved, and their stability analyzed, by converting them into first-order form using a block matrix (matrix of matrices). For example, suppose we have the second-order equation

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

with the variable vector $x$ being $n \times 1$ and $A$ and $B$ being $n \times n$ . This can be stacked in the form

{\begin{bmatrix}\mathbf {x} _{t}\\\mathbf {x} _{t-1}\\\end{bmatrix}}={\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {I} &\mathbf {0} \\\end{bmatrix}}{\begin{bmatrix}\mathbf {x} _{t-1}\\\mathbf {x} _{t-2}\end{bmatrix}}

where $I$ is the $n \times n$ identity matrix and $0$ is the $n \times n$ zero matrix. Then denoting the $2 n \times 1$ stacked vector of current and once-lagged variables as $z t$ and the $2 n \times 2 n$ block matrix as $L$ , we have as before the solution

\mathbf {z} _{t}=\mathbf {L} ^{t}\mathbf {z} _{0}

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.

Nonlinear matrix difference equations: Riccati equations

In linear-quadratic-Gaussian control, there arises a nonlinear matrix equation for the reverse evolution of a current-and-future-cost matrix, denoted below as $H$ . This equation is called a discrete dynamic Riccati equation, and it arises when a variable vector evolving according to a linear matrix difference equation is controlled by manipulating an exogenous vector in order to optimize a quadratic cost function. This Riccati equation assumes the following, or a similar, form:

\mathbf {H} _{t-1}=\mathbf {K} +\mathbf {A} '\mathbf {H} _{t}\mathbf {A} -\mathbf {A} '\mathbf {H} _{t}\mathbf {C} \left[\mathbf {C} '\mathbf {H} _{t}\mathbf {C} +\mathbf {R} \right]^{-1}\mathbf {C} '\mathbf {H} _{t}\mathbf {A}

where $H$ , $K$ , and $A$ are $n \times n$ , $C$ is $n \times k$ , $R$ is $k \times k$ , $n$ is the number of elements in the vector to be controlled, and $k$ is the number of elements in the control vector. The parameter matrices $A$ and $C$ are from the linear equation, and the parameter matrices $K$ and $R$ are from the quadratic cost function. See here for details.

In general this equation cannot be solved analytically for $H t$ in terms of $t$ ; rather, the sequence of values for $H t$ is found by iterating the Riccati equation. However, it has been shown^[3] that this Riccati equation can be solved analytically if $R = 0$ and $n = k + 1$ , by reducing it to a scalar rational difference equation; moreover, for any $k$ and $n$ if the transition matrix $A$ is nonsingular then the Riccati equation can be solved analytically in terms of the eigenvalues of a matrix, although these may need to be found numerically.^[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. See also Stochastic control § Discrete time.

A related Riccati equation^[5] is

\mathbf {X} _{t+1}=-\left[\mathbf {E} +\mathbf {B} \mathbf {X} _{t}\right]\left[\mathbf {C} +\mathbf {A} \mathbf {X} _{t}\right]^{-1}

in which the matrices $X, A, B, C, E$ are all $n \times n$ . This equation can be solved explicitly. Suppose $\mathbf {X} _{t}=\mathbf {N} _{t}\mathbf {D} _{t}^{-1},$ which certainly holds for $t = 0$ with $N 0 = X 0$ and with $D 0 = I$ . Then using this in the difference equation yields

{\begin{aligned}\mathbf {X} _{t+1}&=-\left[\mathbf {E} +\mathbf {BN} _{t}\mathbf {D} _{t}^{-1}\right]\mathbf {D} _{t}\mathbf {D} _{t}^{-1}\left[\mathbf {C} +\mathbf {AN} _{t}\mathbf {D} _{t}^{-1}\right]^{-1}\\&=-\left[\mathbf {ED} _{t}+\mathbf {BN} _{t}\right]\left[\left[\mathbf {C} +\mathbf {AN} _{t}\mathbf {D} _{t}^{-1}\right]\mathbf {D} _{t}\right]^{-1}\\&=-\left[\mathbf {ED} _{t}+\mathbf {BN} _{t}\right]\left[\mathbf {CD} _{t}+\mathbf {AN} _{t}\right]^{-1}\\&=\mathbf {N} _{t+1}\mathbf {D} _{t+1}^{-1}\end{aligned}}

so by induction the form $\mathbf {X} _{t}=\mathbf {N} _{t}\mathbf {D} _{t}^{-1}$ holds for all $t$ . Then the evolution of $N$ and $D$ can be written as

{\begin{bmatrix}\mathbf {N} _{t+1}\\\mathbf {D} _{t+1}\end{bmatrix}}={\begin{bmatrix}-\mathbf {B} &-\mathbf {E} \\\mathbf {A} &\mathbf {C} \end{bmatrix}}{\begin{bmatrix}\mathbf {N} _{t}\\\mathbf {D} _{t}\end{bmatrix}}\equiv \mathbf {J} {\begin{bmatrix}\mathbf {N} _{t}\\\mathbf {D} _{t}\end{bmatrix}}

Thus by induction

{\begin{bmatrix}\mathbf {N} _{t}\\\mathbf {D} _{t}\end{bmatrix}}=\mathbf {J} ^{t}{\begin{bmatrix}\mathbf {N} _{0}\\\mathbf {D} _{0}\end{bmatrix}}

References

^ 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.
^ Balvers, Ronald J.; Mitchell, Douglas W. (2007). "Reducing the dimensionality of linear quadratic control problems" (PDF). Journal of Economic Dynamics and Control. 31 (1): 141–159. doi:10.1016/j.jedc.2005.09.013. S2CID 121354131.
^ Vaughan, D. R. (1970). "A nonrecursive algebraic solution for the discrete Riccati equation". IEEE Transactions on Automatic Control. 15 (5): 597–599. doi:10.1109/TAC.1970.1099549.
^ Martin, C. F.; Ammar, G. (1991). "The geometry of the matrix Riccati equation and associated eigenvalue method". In Bittani; Laub; Willems (eds.). The Riccati Equation. Springer-Verlag. doi:10.1007/978-3-642-58223-3_5. ISBN 978-3-642-63508-3.

[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.

[3] Balvers, Ronald J.; Mitchell, Douglas W. (2007). "Reducing the dimensionality of linear quadratic control problems" (PDF). Journal of Economic Dynamics and Control. 31 (1): 141–159. doi:10.1016/j.jedc.2005.09.013. S2CID 121354131.

[4] Vaughan, D. R. (1970). "A nonrecursive algebraic solution for the discrete Riccati equation". IEEE Transactions on Automatic Control. 15 (5): 597–599. doi:10.1109/TAC.1970.1099549.

[5] Martin, C. F.; Ammar, G. (1991). "The geometry of the matrix Riccati equation and associated eigenvalue method". In Bittani; Laub; Willems (eds.). The Riccati Equation. Springer-Verlag. doi:10.1007/978-3-642-58223-3_5. ISBN 978-3-642-63508-3.

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