LINEAR AND RICCATI MATRIX EQUATIONS

In this paper we find exact solutions for linear ordinary differential equations 
of any order when they are given in matrix form, as well as for classes of Riccati 
matrix equations with two or three arbitrary matrix coefficients. Other nonlinear systems 
of triangular form are also solved completely.

dependent variables equal to the order.Letting each successive one be the derivative of its predecessor, the number of equations will be the same as the order.This can then be put in matrix form.
For example, the linear equation of order n x (n) ao(t)x a (t)x' + is equivalent to the system X I X 2 =X 3 X 2 X' a0xI + a x 2 + n 1 (n-l) + a (t)x (1.1) n-1 + an_iX n (I .2) which can be written as X' =AX where X (x l,x 2 x n) and A We will also treat Riccati matrix equations with two or three arbitrary coeffi- cients and some other nonlinear systems.
For convenience we will assume that all matrices are nxn with elements which are continuous functions of the same variable throughout this section.
The general solution of the linear equation where y is a vector.We solve the first equation Yl II Using y t (x)y + t (x)y 2 etc.
In each instance we we then solve the next equation Y2 12 22 solve a simple linear first-order equation and in each case we find all solutions.Take n linearly independent solutions of (2.8) and write them as n columns.This matrix, called a fundamental matrix of solutions, is a solution of the associated matrix equation (2.7).In the remainder of this section the independent variable x will be under- stood but not written.Other results for triangular matrices include THEOREM 2.2.If S is invertible, Q is nonsingular and S-IQs is triangular, then the equation SX' QSX + T (2.9) is solvable by quadratures for any matrix T.
PROOF.Let S-1QS E, a triangular matrix.Multiplying (2.9) by S -1 we get X' EX S-IT.
First we can solve X' =EX by Theorem (2.1).
COROLLARY 2.4.If A is invertible B is nonsingular and A-1B is trinagular, we can solve AX' BX.
(2.13) We will say a matrix is continuous if each of its elements is a continuous function.
In the next theorem we assume the matrix is continuous.
We now state and prove the main result of this paper.THEOREM 2.5.Let F be an invertible, continuous matrix.Then the equation is solvable by quadratures. PROOF.

The system
We begin with the following.
Let T be a nonsingular triangular matrix and let S be a nonsingular matrix.PROOF OF LEM 2.6.First we note that X' TX-S.

Multiply (2.19) by
(2.16) The solution of (2.16) can be obtained from the solutions of Equation (2.7) by Lagrange's variation-of-constants method referred to in the proof of Theorem (2.2).Each solution of (2.16) yields a unique solution of the system (2.15), since T is non-singular, by use of the relation Y TX-S. (2.17) This completes the proof.
For statement of the next lemma let us assume momentarialy that TU-B is nonsingular for some matrices U and B.
LEMMA 2.7.For any matrices A, B, U, V and W the pair of matrices X UV + AW, Y U BV W is a solution of the system (2.15) if and only if: {U(TU-B)-Iu U(TU-B)-Is + [A + U(TU-B)-I(I-TA)]W}' [I + B(TU-B)-I]u + B(TU-B)-Is + [I + B(TU-B)-I(I-TA)]W. (2.18) Here is the identity matrix.
PROOF OF LEMMA (2.24) and substitute its value in (2.21).
We then see that the equation (2.18) is equivalent to (2.23).This completes the proof.
PROOF OF THEOREM 2.5.By the foregoing we may assume that F is not triangular.
As usual ZI will denote the determinant of Z and stands for the identity.
"Since F is nonsingular there is a nonsingular A such that FA is nonsingular and a lower triangular T (tij) such that TIA1 is also nonsingular.
This is accomplished by imposing nonvanishing restrictions on ITA-II aua FA-I as in the below.
Define P by the equation (FAI-I) (TIAI-I) (2.25) Now since F # T and P is nonsingular, F-PT is nonzero and in fact the coefficient of tll is nonzero in F-PTII.Accordingly, F-PT can be made non- singular by restricting t II" Let C 1 be such that CI-AI, TICI-I are nonsingular.The nonsingularity of (I-P) (F-PTI)C 1 can be obtained from that of F-PT by restricting an element of C I.
At this point we assume that all choices have been made for the elements of AI, TU-B and simplify to get U B)-Iu (TU-B)-Is (TU-B) {I-TA)W.{2.19) summary, the nonsingularity of FAI-I TIAI-I is obtained by restricting a single element of AI; nonsingularity of F-PT 1 by restricting tll nonsingularity of CI-Al, TICI-I and (I-P)-(F-PTI)C by restricting elements of C and finally in the last two cases by use of elements of F-PT I.That this is possible can be seen by