MATRICES 
§ 63] 
103 
summed for i, t, r, s, h = i, . . . . , n. Applying first 
(20) and afterward (6), we get 
Pkj—^ Tjslirsh^hk^r TjsPhs^hk 
r, s, h 
Write Ikh for \kk, and t s j for t> Let T be the matrix 
having t s j as the element in the 5th row and 7th column. 
By (20), htjilu = o or 1 according as j^t or j — t. Hence 
T~ 1 is the matrix having l u as the element in the ith 
row and /th column. Then pij = 'ShhPhstsj gives 
Ri = T~*R t T, S'x = T~ x S x T, 
the second being derived similarly by using (7) instead 
of (6). Thus, if co is an indeterminate, 
R' x —ul = T~ 1 {R x —wI)T, 5*—co/=r _I (5»—co/)T. 
Passing to determinants, we get 
j R x — c0/ | == | R x —c0/ j , | S x col | = | S x col | . 
Theorem. Each characteristic determinant of an 
element x of an algebra, not necessarily associative, over a 
field F, is invariant under every linear transformation of 
units with coefficients in F. The same is therefore true of 
their constant terms A(x) and A'{x). 
63. Lemma on matrices. If a lt a n are the 
roots of the characteristic equation /(co)=o of an n-rowed 
square matrix m whose elements belong to a field F, and if 
g{co) is any polynomial with coefficients in F, then the 
roots of the characteristic equation of the matrix* g{m) are 
g(cti), . . . . , g(a n ). 
* With the term cl if the constant term of g(w) is c.