degree represents scalars, which constitute an ordinary or quadratic matrix; a sum of indeterminate products of the third degree represents 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
being a product of and may be regarded as a product of and by a principle already stated. Now if denotes a sum of indeterminate products, of second degree, say etc., we may write
This is like a quantity of the first degree, and it is a homogeneous linear function of It is easy to see that the most general form of such a function may be expressed in this way. An equation like
represents equations in ordinary algebra, in which variables are expressed as linear functions of others by means of coefficients.
The internal product of two indeterminate products may be defined by the equation
This defines the internal product of matrices, as
This product evidently gives a matrix, the operation of which is equivalent to the successive operations of and i.e.,
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
This defines a multiplication of matrices denoted by the same symbol, as
This multiplication, which is associative and commutative, is of great importance in the theory of determinants. In fact,