XXIX 



variants and covariants was rapidly brought within the range of 

 students through Salmon's ' Lessons on Higher Algebra,' dedicated 

 by the author to Cayley and Sylvester. 



Another subject, of which he must be regarded as the creator, is 

 the theory of matrices. His first memoir* upon this theory, 

 " wherein," to quote Sylvester, f " he may be said to have laid the 

 foundation stone of multiple quantity," was published in 1858. A 

 couple of isolated results had been obtained by Hamilton in 1852 

 through the methods of quaternions ; but they were unknown to 

 Cayley at the time of his memoir, and, owing to the connection in 

 which they occur, they have an entirely detached aspect. 



A matrix may initially be defined as a symbol of linear operation ; 

 thus, when the equations 



X = ax + by + cz, Y = a'x + b'y + c'z, Z = a"x + V'y + c"z 

 are expressed in the form 



(X, Y, Z) = ( a , b , c %x, y, z) = M(or, y, z), 



the symbol M is a matrix. Cayley was the first to discuss the theory 

 of such symbols as subjects of functional operation and to dispense 

 with the hitherto regular return at each stage to the equations of 

 substitution in which the symbol first arises ; in fact, he replaces the 

 notion of substitutional operation by the notion of a new class of 

 quantity. 



Matrices (being of the same order or dimension) can be added like 

 ordinary algebraical quantities ; as regards multiplication, they are 

 subject to the associative law, but not to the commutative law. 

 Hence powers of a matrix (positive and negative, integral and frac- 

 tional) can be obtained, and likewise algebraical functions of a 

 matrix. It also follows that two general matrices are not convertible, 

 that is, LM is riot the same as ML save under special conditions, and 

 it is a part of the theory to find the most general matrix convertible 

 with a given matrix. The expression of this convertible matrix can 

 be deduced by means of the fundamental equation which every matrix 

 satisfies, viz., an algebraical equation of its own order, the coefficient 

 of the highest term being unity, and the last term being the determi- 

 nant of the constants in the matrix. All these results were given by 

 Cayley in his initial memoir ; and, at the same time, they were 

 applied by him to obtain the most general automorphic linear trans- 

 formation of a bipartite quadric function, a problem which is the 

 generalisation of that which requires the most general (orthogonal) 



* ' C. M. P.,' vol. 2, No. 152 ; ' Phil Trans.' (1858), pp. 1737. 

 t ' Amer. Journ. Math.,' vol. 6 (1884), p. 271. 



