EQUIVALENT MATRICES 
T 73 
§ 98] 
equivalent to a matrix d' whose first element is not zero 
and whose further elements in the first row and first 
column are all zero. If the matrix obtained from d f 
by deleting the first row and first column is not zero, we 
may apply to it the result just proved for d. Repetitions 
of this argument show that d is equivalent to a diagonal 
matrix whose elements outside the main diagonal are all 
zero, while those in the diagonal are g XI , . . . . , g nn , each 
of the first r of which are 5^0 and the last n—r are all 
zero (1 <r^n). . Denote this matrix by (g II} .... , gnn) 
and call r its rank. 
If g IX is not both a right and a left divisor of all the 
remaining gjj^o, suppose to fix the ideas that g xl is 
not a left divisor of gu^o. We add the elements of the 
fth row to those of the first row and by (i) obtain an 
equivalent matrix having gu as the fth element of the first 
row. Then by the first part of the proof of Lemma 2 
we obtain an equivalent matrix whose first element g' IX is 
not zero and has a norm numerically <N{g XI ). As 
before we can find an equivalent diagonal matrix whose 
first element is After a finite number of repetitions 
of this process, we reach a diagonal matrix {h It , . , 
h nn ) in which h xl is not zero and is both a right and a left 
divisor of each ha. Treating similarly the matrix 
(,h 22 , . . . . , h nv ), we obtain an equivalent matrix 
(l 22 , . ... , Inn) in which 1 22 is not zero and is both a 
right and a left (jivisor of each la. Morevoer, h xl is both a 
right and a left divisor of l 22 , since they are 
linear combinations of h 22 , . ... , h nn with coefficients 
in S. Proceeding similarly, we obtain the 
Theorem. Every matrix d of rank r> o, whose 
elements belong to a maximal set S of elements of a division