68 PROCEEDINGS OF THE AMERICAN ACADEMY. 



1 m 



SA = ~y a!i = SA* 

 Let 



ft (A) = A m + Pl A" 1 - 1 + . . . + p m _ 2 A 2 + p m _ x A = 



be the equation of lowest order in the first and higher powers of 



A = 2- a «' e »» 

 i 



and let the roots of the scalar equation 



ft (x) = x m + p x x' 1 '- 1 + . . . + p m _ 2 x + p m _ x x = 



be 0, Xi , x 2 , . . . x r} respectively of multiplicity v, v t , y 2 , . . . v r . 

 Either ^4"" = 0, or jt^, p», etc., are not all zero, in which case r ^> 0. 

 For ,4'" 4= 0, and thus r > 0, if we put 



^- i -(^T(^T---(^)' v > 



I have shown in the paper above referred to that the number i(A) of 

 the system is idempotent.f Let the polynomial (x) be defined by the 

 equation 



x®(x) = f(x)t(x). 



Then © (x) is linearly in ar" -1 , X 2 ", etc. Therefore, (A) is a number 

 of the system. Further, since f (A) is idempotent, 



A®(A) = i(A)t(A) = t(A). 



* In my first extension of the quarternion scalar function to matrices in general, 

 referred to in the note p. 69, I defined the scalar function S A of any number 



A = %.%.a..e.. 



* .1 •) '3 

 1 1 



of the matrix, or quadrate of order m, as above, namely, as equal to 



1 Jit 



- %.a... 

 n j « " 



t Loc. cit., p. 524. 



