112 MULTIPLE ALGEBRA. 



degree represents n 2 scalars, which constitute an ordinary or quadratic 

 matrix ; a sum of indeterminate products of the third degree repre- 

 sents 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 



a(\.p) 



being a product of a, X, and p, may be regarded as a product of a|X 

 and p, by a principle already stated. Now if 3? denotes a sum of 

 indeterminate products, of second degree, say a|X + /3|/z + etc., we 



may write 



3>.p 



for a(X.p)+/3(ju ./o) + etc. 



This is like p, a quantity of the first degree, and it is a homogeneous 

 linear function of p. It is easy to see that the most general form 

 of such a function may be expressed in this way. An equation like 



<T = 3>.p 



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 



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 3> 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, 



n 



