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. Occasionally, the time-varying entity may itself be a matrix instead of a vector. 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 matrix difference equations 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 y₀, which is the initial value of the vector y and which must be known in order to find the solution:

y₁ = Ay₀,

y₂ = Ay₁ = AAy₀ = A²y₀,

y₃ = Ay₂ = AA²y₀ = A³y₀,

and so forth. By induction, we obtain the solution in terms of t:

y_t = A^ty₀ = PD^tP ^{− 1}y₀,

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 y₁. 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 y₁ by itself must have a solution involving those same eigenvalues. This description intuitively motivates the equation of evolution of y₁, 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^tz₀.

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 evolution backwards through time 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 to be controlled by manipulating an exogenous vector in order to optimize a quadratic cost function. This Riccati equation assumes the following form or a similar form:

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

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.

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.

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 $X_t = N_tD_t^{-1}$, 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

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

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

= − (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

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

Thus

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

References

1. ^ Cull, Paul; Flahive, Mary; and Robson, Robbie. Difference Equations: From Rabbits to Chaos, Springer, 2005, chapter 7; ISBN 0-387-23234-6.
2. ^ Chiang, Alpha C., Fundamental Methods of Mathematical Economics, third edition, McGraw-Hill, 1984: 608–612.
3. ^ 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.
4. ^ Vaughan, D. R., "A nonrecursive algebraic solution for the discrete Riccati equation," IEEE Transactions on Automatic Control 15, 1970, 597-599.
5. ^ 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.

See also

• Matrix differential equation
• Difference equation
• Dynamical system
• Matrix Riccati equation#Mathematical description of the problem and solution