Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/112

of many remarkable papers on the same subject (which might be more definitely expressed as the algebra of matrices) in various foreign journals.

It is not an accident that this century has seen the rise of multiple algebra. The course of the development of ideas in algebra and in geometry, although in the main independent of any aid from this source, has nevertheless to a very large extent been of a character which can only find its natural expression in multiple algebra.

Our Modern Higher Algebra is especially occupied with the theory of linear transformations. Now what are the first notions which we meet in this theory? We have a set of $n$ variables, say $x,y,z,$ and another set, say $x',y',z',$ which are homogeneous linear functions of the first, and therefore expressible in terms of them by means of a block of $n^{2}$ coefficients. Here the quantities occur by sets, and invite the notations of multiple algebra. It was in fact shown by Grassmann in his first Ausdehnungslehre and by Cauchy nine years later, that the notations of multiple algebra afford a natural key to the subject of elimination.

Now I do not merely mean that we may save a little time or space by writing perhaps $\rho$ for $x,y$ and $z;$ $\rho '$ for $x',y'$ and $z';$ and $\Phi$ for a block of $n^{2}$ quantitiea But I mean that the subject as usually treated under the title of determinants has a stunted and misdirected development on account of the limitations of single algebra. This will appear from a very simple illustration. After a little preliminary matter the student comes generally to a chapter entitled "Multiplication of Determinants," in which he is taught that the product of the determinants of two matrices may be found by performing a somewhat lengthy operation on the two matrices, by which he obtains a third matrix, and then taking the determinant of this. But what significance, what value has this theorem? For aught that appears in the majority of treatises which I have seen, we have only a complicated and lengthy way of performing a simple operation. The real facts of the case may be stated as follows:

Suppose the set of $n$ quantities $\rho '$ to be derived from the set $\rho$ by the matrix $\Phi ,$ which we may express by

$\rho '=\Phi .\rho ;$

and suppose the set $\rho ''$ to be derived from the set $\rho '$ by the matrix $\Psi ,$ i.e.,

$\rho ''=\Psi .\rho ',$

and

$\rho ''=\Psi .\Phi .\rho ;$

it is evident that $\rho ''$ can be derived from $\rho$ by the operation of a single matrix, say $\Theta ,$ i.e.,

$\rho ''=\Theta .\rho ,$

so that

$\Theta =\Psi .\Phi .$