74 BIRKIIOFF AND LANGER. 



Then HSifkf) V- s 0, 



n 



i.e. S (pukii'i = 0, for all choices of the set fci, / t - 2 , . . k n . But by 



construction some v, say »,• is not zero. If then the A*'s are chosen so 

 that ki = 0, i 4= jo, fcy =}= 0, it follows that <p%j= for all i. Inasmuch 

 as no column of <f> contains only vanishing elements this involves a 

 contradiction. Hence the hypothesis | <f> (.r) | =0 is not tenable, and 

 the$ in question fulfills condition (ii), i.e. | <J>(.r) | =f= 0. 



By a change of variable, then, equation (19) can be given the form 

 Y'' = \R\ + B\ Y ■ , where the matrix B, being given by the relation 

 B — <£"' B$> — $ _1 $', is a matrix of continuous functions. Supposing 

 this to have been done we may drop the dashes and consider the equa- 

 tion in the form 



(24) Y'(x)- = [R(x)\ + B(x)\ IXr)-. 



Since we have found that the Y(x) ■ of (20) may be any one of the n 

 vectors l'i(.r)- obtained by replacing T(x) by I\(.r) the further dis- 

 cussion might well be carried through for each of these vectors indi- 

 vidually. However, if the matrix E(x) is defined by the relation 



E(x)^(8 ij e^ x) ) > 



it is readily seen that the matrices Pi(x) can be chosen so that the f 

 column of the matrix 



(25) Y(x) E jPoW + i P 1 (x) +. . . | E{x) 



is precisely the general column of the vector Y,-(x)- Hence all cases 

 are simultaneously treated by the consideration of those formal solu- 

 tions of the matrix equation 



(2G) T(.r)= lR(x)\ + B(x)} Y(x) 



which have the form (25). 



Inasmuch as E'(x) = (\y,-(x) 8< } - e xr J (s) ) = \R(x) E(x), the formal 

 substitution of (25) in (26) gives the identity 



