Matrix difference equation

A matrix difference equation^[1]^[2] 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,

x_t = Ax_{t-1} + Bx_{t-2}

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

x_{t+2} = Ax_{t+1} + Bx_{t}

or as

x_n = Ax_{n-1} + Bx_{n-2}

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

Non-homogeneous first-order case and the steady state

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

x_{t}=Ax_{t-1}+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

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

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

[x_{t}-x^{*}]=A[x_{t-1}-x^{*}].

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:

y_1=Ay_0,

y_2=Ay_1=AAy_0 = A^{2}y_0,

y_3=Ay_2=AA^{2}y_0=A^{3}y_0,

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

y_{t}=A^{t}y_{0}.

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

y_{t}=PD^{t}P^{-1}y_{0},

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: $A^t$ 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. For example, suppose we have the second-order equation

x_t = Ax_{t-1} + Bx_{t-2}

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

\begin{pmatrix}x_t \\ x_{t-1} \\ \end{pmatrix} = \begin{pmatrix} \text{A} & \text{B} \\ \text{I} & 0 \\ \end{pmatrix} \begin{pmatrix} x_{t-1} \\ x_{t-2} \end{pmatrix},

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

z_t = L^{t} 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:

H_{t-1}=K+A'H_{t}A-A'H_{t}C(C'H_{t}C+R)^{-1}C'H_{t}A,

where H, K, and A are n×n, C is n×k, R is k×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 was shown in ^[3] that this Riccati equation can be solved analytically if R is the zero matrix 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

X_{t+1} = -(E+BX_t)(C+AX_t)^{-1}

in which the matrices X, A, B, C, and E are all n×n. This equation can be solved explicitly. Suppose $X_t = N_t D_t^{-1}$ , which certainly holds for t=0 with N₀ = X₀ and with D₀ equal to the identity matrix. Then using this in the difference equation yields

X_{t+1}=-(E+BN_tD_t^{-1})D_tD_t^{-1}(C+AN_tD_t^{-1})^{-1}

=-(ED_t+BN_t)[(C+AN_t D_t^{-1})D_t]^{-1}

=-(ED_t+BN_t)[CD_t+AN_t]^{-1}

=N_{t+1}D_{t+1}^{-1},

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

{\begin{pmatrix}N_{{t+1}}\\D_{{t+1}}\end{pmatrix}}={\begin{pmatrix}-B&-E\\A&C\end{pmatrix}}{\begin{pmatrix}N_{t}\\D_{t}\end{pmatrix}}\equiv J{\begin{pmatrix}N_{t}\\D_{t}\end{pmatrix}}.

Thus

\begin{pmatrix} N_t \\ D_t \end{pmatrix} = J^{t} \begin{pmatrix} N_0 \\ D_0 \end{pmatrix}.

References

↑ Cull, Paul; Flahive, Mary; and Robson, Robbie. Difference Equations: From Rabbits to Chaos, Springer, 2005, chapter 7; ISBN 0-387-23234-6.
↑ Chiang, Alpha C., Fundamental Methods of Mathematical Economics, third edition, McGraw-Hill, 1984: 608–612.
↑ Balvers, Ronald J., and Mitchell, Douglas W., "Reducing the dimensionality of linear quadratic control problems," Journal of Economic Dynamics and Control 31, 2007, 141–159.
↑ Vaughan, D. R., "A nonrecursive algebraic solution for the discrete Riccati equation," IEEE Transactions on Automatic Control 15, 1970, 597-599.
↑ Martin, C. F., and Ammar, G., "The geometry of the matrix Riccati equation and associated eigenvalue method," in Bittani, Laub, and Willems (eds.), The Riccati Equation, Springer-Verlag, 1991.

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