A71 Upper Limit for the Value of a Determinant. 333 
or, say, O2, 0^, Og, the second of which has for its adjugate 
— h^a^ 
— c^a^ 
— d^a^ 
h^a^ 
a^a"" 
-d^a'^ 
c^a^ 
c^a^ 
di:a^ 
al,a^ 
■h^a^ 
d^a^ 
■c^a^ 
b'Za^ 
a^a"^ 
and for its attained upper hmit (Ua^)^. 
18. Bringing together the results of §§ 7, 17, we note that a maximum 
determinant with complex elements is an ant-orthogonant only when the 
elements are equimodular, and that a maximum determinant with real 
elements is always an orthogonant. In the latter case, however, it has to 
be noticed that if the only elements permissible be +1 and — 1 the 
distinction between orthogonant and ant-orthogonant disappears, because 
then 
a;., 
a,.g 
A. 
Cape Town, 
Dec. 28, 1908. 
[19. Added 21/1/09.] As the last step of the reasoning in §5 may not 
carry conviction to some, I append, on Mr. Hough's suggestion, another 
proof of the most direct and simple character : — 
Theorem. — The row-by-row product of two determinants whose corre- 
sponding elements are complex conjugates is not greater than its own 
principal diagonal term. 
Proof. — Let the two determinants and their product be 
(aa') {ah') (ac') 
a^ 
&2 &3 
a', a 2 a. 
{ha') {hh') {he') 
{ca') {ch') {cc') 
Then in the first place it is clear that 
{aa') {ah') 
^ {aa'){bh') 
(I.) 
{ha') {hh') 
because {ah'), {ha') are complex conjugates. In the second place, from a 
well-known property of determinants we have 
{aa') {ah') {ac') 
{ha') {hh') {he') 
{ca') {ch') {cc') 
.-. by (1.) 
and again by (I.) 
{aa') {ah') 
{ha') {hh') 
{aa') {ah') 
{ca') {ch') 
{aa') {ah') 
{ha') {hh') 
{aa') {ac') 
{ha') {he') 
{aa') {etc') 
{ca') {cc') 
^{aa'), 
[aa') {ac') 
{ca') {cc') 
-^{aa'), 
:j> {aa'){hh'){cc'). 
22 
(II.) 
