Vol. 8, 1922 
MATHEMATICS: F.L.HITCHCOCK 
79 
e{x) = AixBi + A2XB2 + . . . + A^xBh = C. (2) 
In 1884 Sylvester attacked this equation with characteristic energy.^ 
He pointed out that, since each element of the matrix 6{x) is a linear 
function of the elements of x — the equation (2) being thus equivalent to 
linear equations of ordinary algebra — it follows d has the properties 
of a matrix of order A^^ and satisfies a Hamilton-Cayley equation 
(?" - mi^-i + W26l"-2 - . . . + i-lTrnJ = 0 (3) 
where n = N^ and where I denotes the matrix of order whose elements 
lik are unity when subscripts are equal, otherwise zero. If we know the 
coefficients nii . . . m„ we can solve (2), as is evident on multiplying (2) 
and (3) through by d~^. We can find all the coefficients if we know the 
rule for finding m„ which is the determinant of the matrix d; for the left 
side of (3) may be found by forming the determinant oi gl — B and after- 
wards putting S in place of g. To Sylvester, solution of the equation (2) 
was equivalent to finding, in terms of the matrices Ai . . . Aj^ and Bi . . . 
the determinant of order belonging to the operation B. 
His general method consists in setting up the two matrices of order 
N and of extent (etendue) h 
XiAi -f X2A2 + . . . + XhAf,, xiBi + X2B2 + . . . + XhBh 
where Xi, X2, etc. are scalar variables. By forming the determinants of 
these two matrices we have two quantics in the variables. He shows that 
the required determinant is a contrariant called the nivellant of these two 
quantics, and is an integral but not necessarily a rational function of the 
coefficients of the quantics. These coefficients are regarded as known 
quantities. The formation of the contrariant is in general difficult. 
He works out the solution for several special cases. With regard to the 
general case he remarks^ 
"Pour presenter 1' expression generale de ce determinant pour une matrice 
d'un ordre et d'une etendue quelconques, c'est-a-dire pour r^soudre I'equa- 
tion lineaire en matrices dans toute sa gen^ralite, il faudrait avoir une 
connaissance des proprietes des formes qui va beaucoup au dela des limites 
des facultes humaines, telles qu'elles se sont manifestees jusqu'au temps 
actuel, et qui, dans mon jugement, ne pent appartenir qu'a I'intelligence 
supreme." 
But the indeterminate product of Gibbs was at that time unborn, or 
at least very young. 
In 1904 Maclagan Wedderburn^ gave a solution of (2) by infinite series. 
His method consists essentially in selecting one of the known matrices, 
as Aj, multiplying by Aj^ and into B]^ giving (2) in the form 
X + xl^ix) = Aj^ CB]' = C 
where is similar in form to B but contains h-l terms. Then 
= (1 + C = {I - ^P ' ^ ' )C'. 
