 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 timevarying 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 secondorder 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 firstorder.
Nonhomogeneous firstorder matrix difference equations and the steady state
An example of a nonhomogeneous firstorder 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 x_{t} = x_{t − 1} = x * in the difference equation and solving for x* to obtain
where I is the n×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:
Stability of the firstorder case
The firstorder matrix difference equation [x_{t}  x*] = A[x_{t1}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 firstorder 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. By induction, we obtain the solution in terms of t:
 y_{t} = A^{t}y_{0} = 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 firstorder matrix system
Starting from the ndimensional 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
where the parameters a_{i} are from the characteristic equation of the matrix A:
Thus each individual scalar variable of an ndimensional firstorder 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 higherorder 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 firstorder form using a block matrix. For example, suppose we have the secondorder 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
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 oncelagged 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 secondorder 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 linearquadraticGaussian control, there arises a nonlinear matrix equation for the evolution backwards through time of a currentandfuturecost 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:
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 , which certainly holds for t=0 with N_{0} = X_{0} and with D_{0} equal to the identity matrix. Then using this in the difference equation yields
 = − (ED_{t} + BN_{t})[CD_{t} + AN_{t}] ^{− 1}
so by induction the form holds for all t. Then the evolution of N and D can be written as
Thus
References
 ^ Cull, Paul; Flahive, Mary; and Robson, Robbie. Difference Equations: From Rabbits to Chaos, Springer, 2005, chapter 7; ISBN 0387232346.
 ^ Chiang, Alpha C., Fundamental Methods of Mathematical Economics, third edition, McGrawHill, 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, 597599.
 ^ 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, SpringerVerlag, 1991.
See also
 Matrix differential equation
 Difference equation
 Dynamical system
 Matrix Riccati equation#Mathematical description of the problem and solution
Categories: Linear algebra
 Recurrence relations
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