TRACE, NILPOTENT 
§6 S ] 
105 
the algebra is solvable in F z and has the roots a z , .... , a„. 
Then the roots of the first {or second) characteristic equation 
of g(x) are g(a), . ... , g{a n ). 
For, the first characteristic equation of x is 
\R x — col\=o, which is the characteristic equation of 
matrix R x , and has the roots a z , , a n . Hence 
by §63 with m = R x , the roots of the characteristic 
equation of matrix g{R x ) are g{a), . ... , g{a n ). 
By (i7j) they are the roots of 
I Rg(x) 1 = 0 J 
which is the first characteristic equation of g{x). 
Corollary*. An element x is nilpotent if and only 
if every root of either characteristic equation of x is zero. 
For, if x r = 0 and if there be a root p^o, the corre 
sponding characteristic equation of x r would have the 
root p’Vo, whereas either characteristic equation of the 
element o is evidently co n = o. 
Conversely, if every root of either characteristic 
equation is zero, that equation is evidently co”=o, 
and by the theorem in § 60 x is a root of the latter or of 
its product by co. 
65. Traces, properly nilpotent elements. The sum 
of the diagonal elements of the first matrix R x of x is 
called the (first) trace of and is denoted by t x . 
The first characteristic equation of x is 
| R x -ul l = (-i) w [co w -/*co w - I + . . . . ]=o. 
Hence t x is equal to the sum of the roots, and (§ 62) is 
independent of the choice of the basal units of the 
algebra. 
* This follows at once from the theorem in § 68.