52 PROCEEDINGS OF THE AMERICAN ACADEMY. 



sponding sums, it represents the sum of the given matrices. If no 

 such matrix exists the sum is complex. In that case we write the 

 result as an algebraic sum and do not attempt to express it as a single 

 matrix. If some of the matrices are expressed in point coordinates, 

 the others in hyperplane coordinates we replace those of one kind by 

 their dual forms or at least imagine them so replaced. This amounts 

 to adding elements of the former to their algebraic compliments in 

 the latter and considering the result as a term of the first kind. 



6. A matrix in which the number of rows is equal to or less than 

 the number of columns can be regarded as a product. If the matrix 

 is expressed in point coordinates we call the product progressive, if 

 in hyperplanar coordinates regressive. To multiply two such matrices 

 (of the same kind) the sum of whose rows is equal to or less than n, 

 we write the second matrix under the first to form a single matrix. 

 If one of the factors is complex we apply the process distributively 

 to the separate matrices of the sum. From the definition it is evident 

 that such products are distributive and associative and that the 

 interchange of two points or hyperplanes (according as the product 

 is progressive or regressive) changes the sign of the result. 



If a matrix of r rows vanishes, all the minors of order r in that matrix 

 are zero. There is then a linear relation between the rows of the 

 matrix.^ If the matrix represents a progressive product of r points 

 there is a linear relation between the points of that product and they 

 therefore lie in a space of order less than r. If the matrix represents a 

 regressive product of r hyperplanes they satisfy a linear relation and 

 therefore intersect in a space of order greater than }i-r. If the matrix 

 is not zero the progressive form represents the space containing its 

 factors and the regressive the space common to its factors. 



The most general product is the result of a succession of operations 

 each consisting of multiplying two factors. If the total number of 

 rows in two matrices of the same kind (progressive or regressive) is 

 less than 7i, the two are multiplied together according to the rule 

 already given. If the total number of rows is greater than n, the 

 product as previously defined gives a matrix of more rows than 

 columns. For such a matrix we have no interpretation. In that case 

 we replace each factor by its equivalent in contragredient variables. 

 The total number of rows in the new product is less than 7i and we form 

 the product by the previous method. If the total number of rows 

 is equal to 7i the result is the same whether the matrices are taken in 



6 Bocher, Introduction to Higher Algebra, p. 36. 



