242 Prof. Dixon, On the greatest value of a 
On the greatest value of a determinant whose constituents are 
limited. (Proof of Hadamard’s theorem.) By Prof. A. C. Drxon, 
E.R.S. 
[Received 9 April 1913—Read 28 April 1913. ] 
Ler there be an array ||@,< || of m rows and n columns (n > _m) — 
the constituent in the rth row and sth column being a,s., a complex 
quantity, whose conjugate is D,s. 
Multiply the two arrays || @;s\|, || 6,s || according to the ordinary 
rule, that is, form the determinant C, of order m, which has in its 
ith row and jth column the constituent 
Ci = = WigDje (2, AR = 1, Dicgets Mm). 
C must be positive since it is the sum of products each formed 
with a determinant from ||q@,,|| and the corresponding determinant 
from ||b,s||, that is, products of conjugate complex quantities. 
The theorem to be proved is that C cannot exceed its leading term 
m 
cz, or say IL. 
1 
Assume this for m — 1, and let Oj; be the coefficient of ¢; in C, 
that is, the product of the matrices 
|| @rs |] and || Br || 
with the zth and jth rows left out respectively, with the sign 
(1), 
Thus the construction of Cj; is the same as that of ej, the 
places of ds, b,; being taken by the first minors of the deter- 
minants of |/q@,5|| and |] 6,.||; the value of m is now different, being 
the number of determinants in a matrix of n columns and m-— 1 
rows. 
Since the theorem is true for m—1 we have 
Cx< Wey (@=1,2...m) 
and also 
eeoecevcereeceeceeee eee 
a 
Fi 
