64 BIRKHOFF AND LANGER. 



That • AB = C, and that AB- = C-, are facts readily established by 

 direct reference to the rules for multiplication. A matrix (a) all of 

 whose elements are the same is written ^4-. From the preceding 

 statement it is clear that -AB- = C-. 



Given W a and W b, any two constant matrices for which | \V a | =£ 

 and | W b | =fc 0, then the two corresponding homogeneous vector 

 differential systems 



, fi s (r(*)- =A(x)Y(x)- 



W \w a Y(a)' +W b Y(b)- =0 



and 



~ \-Z'(x)= - >Z(x)A{x) 



Ki} (■Z(a)W a - 1 + -Z(b)W b - l = 



arc said to be adjoint. 



Theorem: The number of linearly independent solutions of system 

 (()) is always equal to the number of linearly independent solutions of 

 system (7). 



Proof: It has been shown that if F(.r) is any matrix solution of the 

 differential equation (1), the most general solutionis Y(x) =Y(x)C. 

 From this, C = Y' l (.v) Y(z), and if the solution Y(x) is a vector 

 Y(x) ■ then C will be a vector C-. 



The general solution of the differential equation in system (6) is, 

 therefore, 



Y(x)- =T(.r)C-, 



and since the substitution of this in the boundary conditions gives 



(8) W a Y(a)C- +W b Y(b)C- = 0, 



Y(x)- is seen to be a solution of system (G) when and only when C- 

 is a solution of the equation (8). Moreover, it is readily seen that a 

 necessary and sufficient condition that a set of solutions of system ((>) 

 be linearly independent is that the corresponding solutions of equation 

 <8) be independent. Setting IF,, F(a) + Wb Y(b) = (p»y), equation 



(8) can be written in the form 



(9) 2 Pik c k = 0, i=l,2,...n. 



