58 BIRKHOFF AND LANGER. 



it follows, upon substituting from equations (1) and (2), that 



~ZY= (- ZA) Y + Z(AY). 

 dx 



Hence ^- ZY = 0, and Z(x) Y(x) =C. Q. E. D. 



dx 



Converse Theorem: If Y(x) is any matrix solution of equation 

 (1), and the matrix Z(x) is defined by the relation Z(x) Y(x) = C, 

 where | C | =(= 0, then Z(x) is a matrix solution of equation (2). 



Proof: We have jLzY=0, 



dx 



dZ dY 



i.e. — Y + Z— = 0. 



dx dx 



In virtue of equation (1), therefore, 



dZ 



— - Y + ZAY = 0, 

 dx 



and it follows, since | T | =}= 0, that 



dZ 

 dx 



Moreover, \Z\= \ Y 1 \ • | C | 4= 0. Q. E. D. 



Any pair of solutions Y(x) and Z(x) of equations (1) and (2) re- 

 spectively which satisfy the relation Z(x) Y(x) = /, are said to be 

 associated solutions. Thus if Y(x) is any matrix solution of equation 

 (1) the associated matrix solution of equation (2) is Z(x) = Y~ l (x). 



A differential equation is said to be self-adjoint when and only when 

 it is identical with its adjoint after interchange of rows and columns. 

 A necessary and sufficient condition that the equation (1) be self- 

 adjoint is readily seen to be that an = — «/j for all i and j. 



