36 



ALM AU SÆ.S5. 



M.-N. Kl. 



^ 12. A theorem of the symmetric differences of a matrix. 



We learlily sec th.it tlic rxprc.s.sirjn (9) for f^;. in thf [irrcf-ding ^ ( 1 1 ) 



is siiiinlv a translV)riiiali(iii ol" the Coiin : 



1) 



£oL ^ß a dß 



to sum, as usual, for (l and ß which here as al)0\'e must be tiiought of as 



indices i'unninu" fi'om 1 to 



•. The elements f)f tlie matrix oi this 



transformation, i. e. of the matrix '^ßa, are the minors of the second order 

 of the matrix of the f's. In full '^ß a can be written: 



(A) 



J-l-2' 1'2 J-12' IS J-12f " - I " 



•5" - 1 "■ I 2 "î" 1 ", 1 3 5>J 



1 )i, II - I 



i. e. : (conjugate to) the adjoint of F of the second class, h may be 

 denoted by [F.y]. (7^ stands for the primary matrix.) 



But is should be emphasized that the matrix of the transformation 

 ^ß(i(fß, where we have to sum for the first index, is the conjugale (trans- 

 posed) ot" this matrix iA), that is, the matrix of the transformation ^ßadß 

 is ([/'J,). = [FA,. 



The two transformations [F\.y and [Fc\ are, of course, different just as 

 F and Fc are. But we can prove that in this case, where the transformed 

 quantities are the c/'s, it does not make any difference, because there is one 



particular set of ( ) quantities with that property that the two matrices 



[F\ and [/\|._, effect the same transformation on it. This particular set is 

 the symmetric differences of the matrix. This theorem, which we now are 

 going to prove, can be expressed in the following form : 



(a) The tivo mat rices which can be formed from the second udiiors of 

 a primarv matrix and from the second minors of the conjugate of this, 

 transform the syunnrtric differences of the primary matrix into the same 

 set of quantities. 



in order to prove this, we must show that the following equation 

 between the two transfoi^mations in question : 



(2) 



^ß a dß = ^tß dß 



holds good for any a, i. e. for any combination of two rows and columns 

 respectively. 



