37 MATHEMATICS, RECENT TERMINOLOGY 



MATHEMATICS, RECENT TERMINOLOGY IN. 633 



This property, in combination with some of those already mentioned 

 leads very simply to the rule for the multiplication of determinants : 

 for example, we have 



f, a 

 ft, a 1 



a, 

 a',0' 



pa. + <rf), p'a + a'0 



from which the law is obvious. The product might also be'expressed, 

 and although it appears less simple, there ia an advantage in expressing 

 it, in the form 



I pa+<ra.',p0 + ff&' I 



| p'o + ff'o', p'/3 + cr'/3' I ' 



If we omit simultaneously any line and any column of a determinant, 

 and with the terms which are left form a determinant, the determi 

 nants so obtained are the first minors of the given determinant. A 

 similar process, but omitting pairs, triads, &c. of lines and columns, gives 

 the second minors, third minors, &e. of the given determinant. But the 

 first minors are the most important, and are sometimes spoken of simply 

 as the minors. 



A determinant 



a,h,g- 

 M,/ 

 ff,f, e 



11,12- 

 21,22 



where the corresponding terms on opposite sides of the dexter diagonal 

 are equal to each other (say rt=sr) is said to be symmetrical. 



But if the terms are equal in magnitude only, but have opposite 

 signs (say rs= sr, this relation not extending to the terms in the 

 diagonal, for which s=r) the determinant is said to be skew ; and if 

 the relation extends to the case >=r, or what is the same thing, if the 

 terms in the diagonal vanish, the determinant is said to be tkoa 

 tymmetrical. Skew determinants have an intimate connection with 

 the functions called PfafBans. 



COMMUTAJTT. The second rule for the construction of a determinant 

 might have been thus stated, viz. for the determinant 



11, 12.. Ire 

 21,22 



write down the expression 



11 



22 



and permute in every possible way the numbers in the first column, 

 prefixing in each case the sign of the arrangement. Then reading off 



rl 



82 



as meaning 



z n. 

 yl . 2 . . . z n 



the sum of all the terms so obtained is in fact the determinant in 

 question. The same result would be obtained by permuting the 

 numbers in the second column instead of those in the first column. 

 And moreover, if the numbers in both columns are permuted, the sign 

 being the sign + + compounded of the signs corresponding to the 

 separate arrangements, the only difference is, that the determinant 

 will be multiplied by the numerical factor 1. 2. 3. . . n. 



If instead of two we have three or more columns, the resulting 

 function is a commutant. But a distinction is to be made according as 

 the number of columns is even or odd. In the former case wo may 

 permute all but one of the columns, and it is indifferent which column 

 ia left unpermuted ; and if all the columns are permuted, the effect is 

 merely to introduce the numerical factor 1.2. 3.... n. In the latter 

 case, if all the columns are permuted, the result is zero, and it is there- 

 fore essential that one column should remain unpermuted ; moreover, 

 different results are obtained according to the column which is left 

 unpermuted, and such column must therefore be distinguished ; this 

 is done by placing above it the mark t. 



PFAFFIAX.- -Suppose that the terms 12, 13, 21, &c., are such that 

 21 = 12, and generally that sr= n, then the Pfaffians 1234, 123456, 

 &c., are denned by means of the equations 



1234 = 12.34 + 13.42 + 14.23, 



123456 = 12.3456 + 13.4562 + 14.5623 + 15.6234 + 16.2345, 

 (where of course 3450 = 34.56 + 35.61 + 36.46, and so for 

 4562, &c.) 



and so on. The functions in question occur in the solution of an 

 important problem (including that of partial differential equations of 

 tho first order and of any degree) known as PfafFs problem, and were 

 named accordingly. 



It may be noticed that a skew symmetrical determinant of any odd 

 order is equal to zero ; but that a skew symmetrical determinant of any 



even order is the square of a Pfaffian, e.g. if 12= -2], &c., as above, 

 then 



0, 12, 13, 14 

 21, 0, 23, 24 

 31,32, 0,34 

 41,42,43, 



= (12.34 + 13.42 + 14.23)". 



_ PERMUTANT. A very simple instance of a permutant is as follows, 

 Tlz - : v m> V 213 , &c., being any quantities whatever, then the permu- 

 tert ((VIM)) denotes the sum 



V 4- V 4- V V TT V 



123^ 23l T V 3 12 V 132 \ 213 V 321 



and in like manner for any number of permutable suffixes, or if instead 

 of a single set of permutable suffixes we have two or more sets of such 

 suffixes. It will be at once obvious how a permutant includes a deter- 

 minant, commutant, or Pfaffian ; thus, if v 123 denotes 0^7, and 

 therefore V 213 = o.,0 1 7 3 , &c., then we have a determinant, so if v ,, 

 denotes o 12 .o 31 where o^ = -a 12 , &c., we have a Pfaffian. 



INTERMUTAKT is a special form of permutant which need not be here 

 further explained. 



CCMULANT. The name has been given to the function which is 

 the numerator or denominator of a continued fraction. Such function 

 may be exhibited (and indeed naturally presents itself) in the form 

 of a determinant, thus the cumulant (abed) or numerator of the 



fractions a + i~j_ ,7 is 



a, I, , 

 -1, b, 1, 

 ,-1, c, 1 



and so for a greater number of terms. The developed expression is 

 abcd + ab + ad+bc+l, which is formed from the produced a b c d by 

 successively omitting each product (cd, be, ab), or set of products 

 (cd, ab) of two consecutive letters ; in like manner the cumulant 

 (abcde) is abcde + abc + abe + ade + a + c + e. 



MATRIX. The term might be used to denote any arrangement of 

 terms, but in a restricted sense it denotes a square or rectangular 

 arrangement of terms, and it is thus employed in the theory of 

 determinants. 



To show further how the notion of a matrix is made use of, it may 

 be remarked that a system of linear equations 



f=o x + b y + c z, 

 i\ = a' x + V y + cf z, 



is in the notation of matrices represented by 



(1, 57. () = (a, b, c ) (x,y,z) 

 I a', b', c 1 



\ a", y, o" 



The corresponding set of equations which give (x, y, z) in terms of 

 ({> f> is represented by, 



(x,y,z) = ( 



a, b, c ))-' 

 a', b', d 

 a", b", c" 



and we have thus the definition of the inverse or reciprocal matrix : it 

 follows from the theory of determinants that the terms of the recipro- 

 cal matrix are the first minor determinants formed out of the original 

 matrix, each of them divided by the determinant formed out of the 

 original matrix ; but in writing down the expression some attention is 

 required with respect to the arrangement and signs of the terms. 



Similar considerations lead to the notion of multiplying or compound- 

 in;/ together two or more matrices. As an instance of such composition, 

 take 



j8 ) = ( po + <ra', 



(p, 



|p', < 



| 



+ <rV,p'/3+<r',8'| 



where it is to be observed that the lines of the first or further compo- 

 nent matrix are compounded with the columns of the second or nearer 

 component matrix to form the lines of the compound matrix. The 

 words further, nearer, are used in reference to a set (x, y) which is, or 

 may be considered to be, understood at the right of each side of tho 

 equation. A matrix may be compounded with itself once or oftener, 

 jiving rise to a positive power of such matrix ; the notion of the nega- 

 tive powers is deducible from that of the inverse or reciprocal matrix, 

 and the same process of generalisation as is employed for powers of a 

 single quantity leads to the notion of the fractional powers of a matrix. 

 As a definition of addition, matrices are added together by the addi- 

 ;ion of their corresponding terms, and as a particular case of the 

 multiplication or composition of matrices we have the multiplication 

 of a matrix by a single .quantity, effected by multiplying by such 

 juantity each term of the matrix ; all these notions together lead to 

 ;he notion of functions of a matrix. 



