§ 58] EQUIVALENCE TO MATRIC ALGEBRA 
97 
tion A is equivalent to the algebra with the elements 
R m and is reciprocal to that with the elements S m . 
If A does not have a modulus, we employ the associa 
tive algebra A* over F with the set of basal units 
u 0 , u I} .... , u n , where the annexed unit u 0 is such 
that 
(10) U 0 tio , UqUi Ui —X, • • « ■ , tz) , 
and hence is the modulus of A*. Write 
(11) X* = £ 0 U 0 +X, Z* = £ 0 U 0 +Z, Z*' = £' 0 u 0 +z', 
where x, z, z' are the elements of A displayed above (3). 
Then 
^oÇo'Mo~\~XÇ 0 ~\~ £ 0 Z TXZ , 
Equating this to 2*, we obtain the transformation 
f0 — £ofo, h — &fo+ ^ ^ irrijkt} 
(k = i, .... , n). 
The matrix of the coefficients of ■ Ç' 0) . . . . , Ç' k is 
Rl*. The latter are the elements of an algebra equivalent 
to A* by Theorem 2. Now x* is in A if £„ = 0. Hence 
the elements x of A are in one-to-one correspondence 
with the matrices 
/0 o . . . . 
PlI .... 
£« p»i • . . •