654 PROCEEDINGS OF THE .AMERICAN ACADEMY. 



The polynomial (f)(p) is termed the "characteristic function" of Ay 

 and <t)(p) = the "characteristic equation" of A. Since, by (67), 

 w S ^ is the sum of the constituents in the principal diagonal of the 

 matrix representing A, it follows that n S A is equal to the sum of the 

 roots of the characteristic equation of A. 



If A is idempotent, the roots of its characteristic equation are and 

 1. Wherefore, if A is idempotent, n S A is equal to the multiplicity of 

 the root 1 of the characteristic equation of A. 



In conformity with the notation employed in § 2, let 



(71) fl ( J) = AJ^ + p,A^-' + . . . + p,-iA = 



be the syzygy of lowest order in powers of A. Then p</) (p) contains 

 fl (p). Whence it follows that n is the maximum number of distinct 

 non-zero roots of the equation fi (p) = 0. Therefore, by theorem III, 



and what was proved p. 636, A is nilpotent if, for some positive integer p, 



'SAP-'^ = (h^ 0,1,2, ...n- 1). 



Conversely, by theorem I, if A is nilpotent, these equations are satisfied 

 for any positive integer p. 



n 



For the scalar functions defined in § 1 of an}- number A—^ ai Ci 



t=i 



of the system {ci, e^, ... e^) I shall write ^\A and 82^ as in § 1 and 

 § 2. The symbol S also is significant when prefixed to any letter de- 

 noting a number of the system {e\, e<i, ... Cm), since any such number 

 belongs to the quadrate €«« (m, t), = 1, 2, . . . n). We have, by (62) 

 and (67), 



(72) Sei= ^L 0„/) (i= 1,2, ... «); 

 and, therefore 



m ■, m n 



(73) S J = 1: a^ei =11 aiQj'\ 



t = l ^^ 1=1 u = l 



Let now 



m m n n 



(74) X= Z ^iCi = JL L L Xidu^'^euv-, 



i=l t = l w=l i)=l 



