TRANSFORMATION OF UNITS 
ioi 
§ 61] 
Let x be an element of any algebra A which need not 
be associative nor have a modulus. The matrices 
R x —wl= (pkj—uôkj) , S x ~œl=(a k j—uôkj) , 
in which 8jj= i, 8kj=o(k9^j), are called the first and 
second characteristic matrices of x, while their determinants 
5(a)) and 8'(to) are called the first and second characteristic 
determinants of x. Thus the first characteristic matrix 
of x is obtained by subtracting to from each diagonal 
element of the first matrix R x of x. 
When A is associative, 5(oj)=o or o)5(o))=o and 
8 / (co)=o or to8'(to)=o are called the first and second 
characteristic equations of x, according as A has or lacks 
a modulus. 
These terms are all relative to the chosen set of basal 
units u 1} . . . . , u„ of A. However, we shall next 
prove that 8 (to) and 8'(to) are independent of the choice 
of the units. 
61. Transformation of units. This concept was 
introduced in § 6. But we now need explicit formulae. 
Let m« be a set of basal units of any 
algebra A, not necessarily associative, over a field F. 
We may introduce as new units any n linearly independ 
ent elements of A: 
n 
where the r# are numbers of F of determinant ¡Ty] ¿¿o. 
Then equations (18) are solvable for the Uj; let the solu 
tion be