Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.
For the subjects of this general heading see the articles ALGEBRA; ALGEBRAIC FORMS; ARITHMETIC; COMBINATORIAL ANALYSIS; DETERMINANTS; EQUATION; FRACTION, CONTINUED; INTERPOLATION; LOGARITHMS; MAGIC SQUARE; PROBABILITY.
Dodgson periodically published mathematical works - An Elementary Treatise on Determinants (1867); Euclid, Book V., proved Algebraically (1874); Euclid and his Modern Rivals (1879), the work on which his reputation as a mathematician largely rests; and Curiosa Mathematica (1888).
The theories of determinants and of symmetric functions and of the algebra of differential operations have an important bearing upon this comparatively new branch of mathematics.
Pp. 8 0 -94, 95112) showed by his calculus of hyper-determinants that an infinite series of such functions might be obtained systematically.
THE Theory Of Determinants.'
So that A breaks u p into a sum of determinants, and we also obtain a theorem for the addition of determinants which have rows in common.
Similarly, by putting one or more of the deleted rows or columns equal to rows or columns which are not deleted, we obtain, with Laplace, a number of identities between products of determinants of complementary orders.
From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of A= (a ll, a22, ï¿½ï¿½ï¿½ ann) and D = (b21, b 22, b nn) may be written as a determinant of order 2n, viz.
We may say that, in the resulting determinant, the element in the ith row and k th column is obtained by multiplying the elements in the kth row of the first determinant severally by the elements in the ith row of the second, and has the expression aklb11+ak2b12+ak3b13ï¿½ï¿½ï¿½ +aknbin, and we obtain other expressions by transforming either or both determinants so as to read by columns as they formerly did by rows.
If we form the product A.D by the theorem for the multiplication of determinants we find that the element in the i th row and k th column of the product is akiAtil+ak2A12 +ï¿½ï¿½ï¿½ +aknAin, the value of which is zero when k is different from i, whilst it has the value A when k=i.
Such determinants are called symmetrical.
Functional determinants were first investigated by Jacobi in a work De Determinantibus Functionalibus.
ï¿½ Oxl d 2x 77n If we have new variables z such that zs=4s(yl, Y2,...yn), we have also z s =1 Y 8(x1, x2,ï¿½ï¿½ï¿½xn), and we may consider the three determinants which i s 7xk, the partial differential coefficient of z i, with regard to k .
Linear Equations.-It is of importance to study the application of the theory of determinants to the solution of a system of linear equations.
Cayley, however, has shown that, whatever be the degrees of the three equations, it is possible to represent the resultant as the quotient of two determinants (Salmon, l.c. p. 89).
References For The Theory Of Determinants.-T.Muir's "List of Writings on Determinants," Quarterly Journal of Mathematics.
Muir, History of the Theory of Determinants (2nd ed., London, 1906).
The fundamental system connected with n quadratic forms consists of (i.) the n forms themselves f i, f2,ï¿½ï¿½ fn, (ii.) the (2) functional determinants (f i, f k) 1, (iii.) the (n 2 1) in variants (f l, fk) 2, (iv.) the (3) forms (f i, (f k, f ni)) 2, each such form remaining unaltered for any permutations of i, k, m.
The germs of the theory of determinants are to be found in the works of Leibnitz; Etienne Bezout utilized them in 1764 for expressing the result obtained by the process of elimination known by his name, and since restated by Arthur Cayley.
His name appears in 1477 in the Register of the Faculty of Arts at St Andrews, among the Determinants or Bachelors of Arts, and in 1479 among the masters of the university.
Besides the conventional use of certain signs as the indications of names of gods, countries, cities, vessels, birds, trees, &c., which, known as " determinants," are the Sumerian signs of the terms in question and were added as a guide for the reader, proper names more particularly continued to be written to a large extent in purely " ideographic " fashion.
It was considered by Biot to have been originally twenty-four, but to have been enlarged to twenty-eight about i ioo B.C., by the addition of determinants for the solstices and equinoxes of that period.
Eight junction stars lie quite close to, seven others are actually identical with, Chinese determinants; 14 and many of these coincidences 9 Sir William Jones, As.
Eighteen Chinese determinants were included in the Arab asterisms, and of these five or six were not nakshatra stars; consequently, they must have been taken directly from the Chinese series.
As a mathematician he occupied himself with many branches of his favourite science, more especially with higher algebra, including the theory of determinants, with the general calculus of symbols, and with the application of analysis to geometry and mechanics.
To a mind imbued with the love of mathematical symmetry the study of determinants had naturally every attraction.
To this, the first elementary treatise on determinants, much of the rapid development of the subject is due.
Given the presence of all the necessary determinants for the development of pigment in a mammal's coat, some or all of the hairs may bear this pigment according to the pattern determinants, or absence of pattern determinants, which the cells of the hair papillae carry.
And this leads to the inquiry as to whether albinoes ever exhibit evidence that they carry the pattern determinants in the absence of those for pigmentation.
For it is to be expected a priori that, since albinoes were derived from pigmented progenitors and may at any time appear, side by side with pigmented brothers, in a litter from pigmented parents, they would be carrying the pattern determinants of some one or other of their pigmented ancestors.
Moreover, the functions b'c" - b"c', b"c - bc", bc' - b'c used in the process are themselves the determinants of the second order l b",c"I?
I b",c"I I We have herein the suggestion of the rule for the derivation of the determinants of the orders 1, 2, 3, 4, &c., each from the preceding one, viz.
Multiplication of two Determinants of the same Order.
Broken up into a sum of (3 3 =) 2 7 determinants, each of which is either of some such form as a, a, a' a' b' a", a", where the term a/3y' is not a term of the a/3y-determinant, and its coefficient(as a determinant with two identical columns) vanishes; or else it is of a form such as t af'y".
- Consider, for simplicity, a determinant of the fifth order, 5= 2+3, and let the top two lines be a, b, c, d, e a', c' d'e', ,, then, if we consider how these elements enter into the determinant, it is at once seen that they enter only through the determinants of the second order a,: b' I, &c., which can be formed by selecting any two columns at pleasure.
Moreover, representing the remaining three lines by a" b" c" d" e" b /r c a, d N, e"' a " c 'N d"N err" it is further seen that the factor which multiplies the determinant formed with any two columns of the first set is the determinant of the third order formed with the complementary three columns of the second set; and it thus appears that the determinant of the fifth order is a sum of all the products of the form ' a b c" d" e" a, b"c"'dN, ear the sign being in each case such that the sign of the term .c"d"'e" obtained from the diagonal elements of the component determinants may be the actual sign of this term in the determinant of the fifth order; for the product written down the sign is obviously +.
A determinant is symmetrical when every two elements symmetrically situated in regard to the dexter diagonal are equal to each other; if they are equal and opposite (that is, if the sum of the two elements be = o), this relation not extending to the diagonal elements themselves, which remain arbitrary, then the determinant is skew; but if the relation does extend to the diagonal terms (that is, if these are each = o), then the determinant is skew symmetrical; thus the determinants a, h, g a, v, - µ 0, v, - h, b, f - v, h, - v, 0, g,f,c c 12, - X, o are respectively symmetrical, skew and skew symmetrical: =0; a,b,c,d a' b' c' d'a" b c d" a, b, c, d a' b' c' d'a", b N' c N' dN,, , c d The theory admits of very extensive algebraic developments, and applications in algebraical geometry and other parts of mathematics.
For further developments of the theory of determinants see Algebraic Forms. (A.
- These functions were originally known as " resultants," a name applied to them by Pierre Simon Laplace, but now replaced by the title " determinants," a name first applied to certain forms of them by Carl Friedrich Gauss.
The germ of the theory of determinants is to be found in the writings of Gottfried Wilhelm Leibnitz (1693), who incidentally discovered certain properties when reducing the eliminant of a system of linear equations.
Determinants were also employed by Etienne Bezout in 1764, but the first connected account of these functions was published in 1772 by Charles Auguste Vandermonde.