312 
ME. H. J. S. SMITH ON SYSTEMS OE LINEAE 
where ||C|| is a prime matrix, it is clear that every minor of ||A]| is a linear function of the 
minors foi inecl from the same horizontal rows of ||B|| ; so that if a and h be the greatest 
common dmsors of any corresponding horizontal rows of minors in those two matrices, 
a is divisible by h. But again, if S be any one of the determinants of ||C||, and s be the 
order of the minors under consideration, any minor of ||B||, after multiplication by 
may be expressed as a linear function of a certain group of the minors taken from the 
same horizontal rows of ||A|. Consequently is divisible by a ; or, since & may have 
any one of a series of values which are relatively prime, d is divisible by a, i. e. l/ = a. 
By combining these results we obtain the theorem. 
“If Vn 5 V«-i 5 Vn- 21 -- - Vi represent the greatest common divisors of all the minors of 
order n, n — 1, ... 1, respectively which can be formed out of a given square matrix, these 
numbers will remain unchanged, when the given matrix is premultiplied by any unit- 
matrix, and post-multiplied by any prime matrix whatsoever.” 
Art. 13. Let S, the determinant of the square matrix 
nxn 
a 
, be a positive number, 
different from zero. It may be shown that by post-multiplication with a properly 
assumed unit Pajl, the matrix 'a\ can be reduced to the form 
(62.) 
j /^15 ^ 1, 
25 ^1,3 • • 
• • ^”l, H 
0 5 ^2 
5 G, 3 • • 
• • ^' 2 , n 
• o 
• o 
5 f^3 • • 
• • r,^n 
0 , 0 
,0 .. 
• • 
Avhere ... are positive numbers, such that and the consti- 
tuents satisfy the inequalities 
(63.) 
This was first observed by Gauss for the case n=2; by Seebee for n=o; and the 
general theorem has been enunciated by M. Heemite*. Its precise statement is 
“ Every matrix of the type nxn is equivalent (by post-multiplication) to one, and only 
one, of the reduced matrices included in the formula (62.).” 
To show this, let i, ^ 2 , n • •• i be the integral and relatively prime numbers which 
satisfy the equations 
G, 1? ^ 1 , l~\~ 25 ^2, 1 ~1~ • • • H,5 1 ^ 
i=2, 3, ... n 
(64.) 
and the inequality 
15 ^1, I 25 ^2, 1 ~1~ • • • ®1, »5 '^n, I 6' 
Then it is evident that, if |[v|| be a unit-matrix of which „ -yg, „ , form the first 
column, the matrix ||«|j X ||'y|| will assume the form 
* CrATJSs, Disq. Arith. art. 213; Seebee, “ Untersuchungen ueber die Eigenschaften der positiven 
temaren quadratischen Eormen” (Alamiheim, 1831), art. 31 ; M. Heemite, Ceelle, vol. sli. p. 192. 
