WILSON: REDUCTION OF REAL DYADICS 175 



/. ={AI +B*+ . . . +H*" 1 ' 13 ) (±-I+B f *+ . . . +GV -1 ) 



= l-<t> p p O) 



with similar expressions for I b , I c . . . . corresponding to 

 each of the roots b, c, . . . . The dyadics I a , I b , . . . are 

 partial idemfactors, their squares are equal to themselves, they 

 are independent, the product of two having different subscripts 

 vanishes (I a I b '= 0) , and the sum of all is the idemfactor I. 

 (d) The dyadic <f> may be written as a sum of terms (p. 28) 



$ = $/ = $ {I a + I b + . . . ) 

 = $ B -f- <S> 6 - + $ c + . . . 



where the product of any two vanishes ($ a $ 6 =0). The dy- 

 adics 



^ a ^a **-* a j • • • j 



are nilpotent, i. e., Z„ = 0. The series of powers (p. 29) 



Z a , Z~ a , . . . , Z a , Z a = 



have increasing nullities, but the change of nullity between two 

 succeeding powers never increases. 



The reduction of $ has thus been simplified to that of nilpo- 

 tent dyadics Z. Beginning with Z v ~ l we may work back through 

 descending powers to Z a and hence to 3> a . We thus find the 

 familiar result that, when expressed in matrical or quadrate 

 form, $ consists of a set of terms along the main diagonal, with 

 at most some terms in the next parallel partial diagonal (called 

 shearing terms, p. 31). 



3. What additional information is obtainable if the dyadic is 

 real? The steps in the proof may be traced one at a time. 



(a 1 ) The equation of least degree must be real since it is 

 unique — a complex equation is equivalent to two. 



(&') The complex roots of A (x) = occur in conjugate pairs 

 of the same multiplicity. Hence if a and b are conjugate imag- 

 inaries, p and q are equal. 



(c') If a and b are conjugate imaginaries, so are 3> — al and 

 $ — bl, and hence so must be I a and I b , for they are obtained by 

 similar real operations applied to conjugate imaginaries. 



