54 



BIRKHOFF AND LANGER. 



are equivalent. It follows from the relations AA~ 1 = I and I A = AI, 

 however, that AA~ l A = IA = AI, i.e., A(A~ l A - I) = 0. 

 Hence if A is any matrix for which | A | :£ 0, it is seen that 



A- 1 A -1=0, i.e., A~ l A = I = AA~\ 



In accordance with the following definitions, namely 



and 



Ja(x) dx = (faM dxj 



a a 



dA(x) (daij(x)\ 



dx 



dx 



a matrix is seen to be integrable or differentiable if and only if this is 



true of each of its elements. It is also clear that if C is a matrix of 



dC 

 constants, then — = 0, while for any product 

 ax 



d dB dA n 



— AB = A— + —B. 



dx dx dx 



Section II. 



The equation Y'(x) = A(x) Y{x)A 



Consider any matrix of functions Y(x) which satisfies (i.e. is a 

 solution of) the equation 



(1) Y'= AY. 



In accordance with the rules for determinants we have 



d\ Y 



dx 



2/n 



2/21 



■Vln 



■yin 



y n \ ■ ■ ■ y nn 



+ 



2/ii 



2/21 



■2/ln 



■ yin 



2/nl 



•2/nn 



4 For the general theory of matrix differential equations see Schlesinger, L., 

 Vorlesungen fiber lineare Different ialgleichungen. Leipzig; B. G. Teubner, 

 1908. 



