Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/113

In the language of multiple algebra $\Theta$ is called the product of $\Psi$ and $\Phi .$ It is of course interesting to see how it is derived from the latter, and it is little more than a schoolboy's exercise to determine this. Now this matrix has the property that its determinant is equal to the products of the determinants of $\Psi$ and $\Phi .$ And this property is all that is generally stated in the books, and the fundamental property, which is all that gives the subject its interest, that $\Theta$ is itself the product of $\Psi$ and $\Phi$ in the language of multiple algebra, i.e., that operating by $\Theta$ is equivalent to operating successively by $\Psi$ and $\Phi ,$ is generally omitted. The chapter on this subject, in most treatises which I have seen, reads very like the play of Hamlet with Hamlet's part left out.

And what is the cause of this omission? Certainly not ignorance of the property in question. The fact that it is occasionally given would be a sufficient bar to this answer. It is because the author fails to see that his real subject is matrices and not determinants. Of course, in a certain sense, the author has a right to choose his subject. But this does not mean that the choice is unimportant, or that it should be determined by chance or by caprice. The problem well put is half solved, as we all know. If one chooses the subject ill, it will develop itself in a cramped manner.

But the case is really much worse than I have stated it. Not only is the true significance of the formation of $\Theta$ from $\Psi$ and $\Phi$ not given, but the student is often not taught to form the matrix which is the product of $\Psi$ and $\Phi ,$ but one which is the product of one of these matrices and the conjugate of the other. Thus the proposition which is proved loses all its simplicity and significance, and must be recast before the instructor can explain its true bearings to the student. This fault has been denounced by Sylvester, and if anyone thinks I make too much of the standpoint from which the subject is viewed, I will refer him to the opening paragraphs of the "Lectures on Universal Algebra" in the sixth volume of the American Journal of Mathematics, where, with a wealth of illustration and an energy of diction which I cannot emulate, the most eloquent of mathematicians expresses his sense of the importance of the substitution of the idea of the matrix for that of the determinant. If then so important, why was the idea of the matrix let slip ? Of course the writers on this subject had it to commence with. One cannot even define a determinant without the idea of a matrix. The simple fact is that in general the writers on this subject have especially developed those ideas which are naturally expressed in simple algebra, and have postponed or slurred over or omitted altogether those ideas which find their natural expression in multiple algebra. But in this subject