Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/128

degree represents $n^{2}$ scalars, which constitute an ordinary or quadratic matrix; a sum of indeterminate products of the third degree represents $n^{3}$ scalars, which constitute a cubic matrix, etc. I shall confine myself to the simplest and most important case, that of quadratic matrices.

An expression of the form

$\alpha (\lambda .\rho )$

being a product of $\alpha ,\lambda ,$ and $\rho ,$ may be regarded as a product of $\alpha \vert \lambda$ and $\rho ,$ by a principle already stated. Now if $\Phi$ denotes a sum of indeterminate products, of second degree, say $\alpha \left\vert \lambda +\beta \right\vert \mu +$ etc., we may write

$\Phi .\rho$

This is like $\rho ,$ a quantity of the first degree, and it is a homogeneous linear function of $\rho .$ It is easy to see that the most general form of such a function may be expressed in this way. An equation like

$\sigma =\Phi .\rho$

represents $n$ equations in ordinary algebra, in which $n$ variables are expressed as linear functions of $n$ others by means of $n^{2}$ coefficients.

The internal product of two indeterminate products may be defined by the equation

$(\alpha \vert \beta ).(\gamma \vert \lambda )=(\beta .\gamma )\alpha \vert \delta .$

This defines the internal product of matrices, as

$\Psi .\Phi .$

This product evidently gives a matrix, the operation of which is equivalent to the successive operations of $\Phi$ and $\Psi ;$ i.e.,

$(\Psi .\Phi ).\rho =\Psi .(\Phi .\rho ).$

We may express this a little more generally by saying that internal multiplication is associative when performed on a series of matrices, or on such a series terminated by a quantity of the first degree.

Another kind of multiplication of binary indeterminate products is that in which the preceding factors are multiplied combinatorially, and also the following. It may be defined by the equation

$(\alpha \vert \lambda )_{\times }^{\times }(\beta \vert \mu )_{\times }^{\times }(\gamma \vert \nu )_{\times }^{\times }=\alpha \times \beta \times \gamma \vert \lambda \times \mu \times \nu .$

This defines a multiplication of matrices denoted by the same symbol, as

$\Phi _{\times }^{\times }\Psi _{\times }^{\times }\Omega ,$$\Phi _{\times }^{\times }\Psi _{\times }^{\times }\Omega _{\times }^{\times }\Theta .$

This multiplication, which is associative and commutative, is of great importance in the theory of determinants. In fact,

${\frac {1}{\vert {\underline {n}}}}\Phi _{\times }^{\times }\!^{n}$