A determinant 1 a,h,g••11, 12 •• h, b, f I or 1 21, 22 , a • .
whero the corresponding terms on opposite sides of the dexter diagonal are equal to each other (say r s=s r) is said to be symmetrical.
But if the terms aro 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 s= r, or what is the same thing, if the terms in the diagonal vanish, the determinant is said to be skew symmetrical. Skew determinants have an intimate connection with the functions called Pfaffians.
Coaienvv er.-The second rule for the construction of a determinant might have been thus stated, viz. for the determinant 11,12..1n 21, 22 nl write clown the expression 11 22 tin and permute in every possible way the numbers in the first column, prefixing in each case the sign of the arrangement. Then reading off r 1 z n.
as meaning 82... 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 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 we may
permute all but one of the columns, and it is indifferent which column is 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.
Pritrrissr.- -Suppose that the terms 12, 13, 21, &e., are such that 21=-12, and generally that sr= -rs, then the Pfaffiane 1234, 123456, &c., are defined 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 3456=39.56+35.64+36.45, 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 the first order and of any degree) known as Pfaff's 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= -21, &c., as above then 0, 12,13, 14 = (12.34 +13.42 + 14.23)a.
21, 0, 23, 24 31, 32, 0, 34 41, 42, 43, 0 PLInft1TANT.-A very simple instance of a permutant is as follows viz. : v„„ v,,,, &c., being any quantities whatever, then the permu. taut denotes the sum