666 MOORE. 



various powers of fx are sums of principal minors of the determinant 

 in which the n's, are replaced by zeros. That is the determinant can 

 be expanded! in the form 



(22) m'" + -liM-"'-' + A,ix~-'-'~ +....+ Ao„, = 



where A\ is the sum of the first minors in which ii has been replaced by 

 zero. These minors are then skew symmetric determinants of odd 

 order and therefore vanish. For the same reason all the y4's with 

 odd subscripts vanish. The coefficient An is the sum of all the second 

 principal minors with fx replaced by zero. This is a skew symmetric 

 determinant of even order and therefore can be expressed as a sum 

 of squares, consequently it is positive. The same is true of all ^'s 

 with even subscripts. Then (22) has only even powers of ix and all 

 the coefficients are positive. Therefore the roots appear in conjugate 

 imaginary pairs. This means for M real the invariant vectors appear 

 in conjugate imaginary pairs. But two conjugate imaginary vectors 

 determine a real plane. Hence we have shown that there are always 

 VI invariant real planes. If equations (21) are not all independent 

 there will be an infinite number of values of n and consequently 

 an infinite number of invariant planes. This corresponds to the case 

 {M^M)-M = \M. 



Now having determined that in every case there is at least one 

 invariant simple plane, A say, we can determine X so that M — \A 

 will be a complex vector lying in a space of 2m — 2 dimensions which 

 must be completely perpendicular to A. To determine X we have 

 the relation 



{M - XA)x{M - \A)x. ..m factors = 

 = MxMx. . .m factors — m\{A>^M>^M . . . {m — 1) factors). 



From which 



X = 



.Jxi/ 



m-\ 



when the exponents indicate cross multiplication. The same reason- 

 ing as used before will show that the space in which M — \A lies 

 is completely perpendicular to A. Thus we have established the 

 theorem: A complex 2-vcctor in a space of 2p dimensions can always be 

 resolved inio the smn of p mvtvaUy perpendicidar planes. If the set of 

 2-^-ectors A, B,— P of {20) are not independent this resolution is not 

 unique. 



