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ALGEBRAIC FORMS

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Originally appearing in Volume V01, Page 626 of the 1911 Encyclopedia Britannica.
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ALGEBRAIC FORMS. The subject-matter of algebraic forms is to a large extent connected with the linear transformation of algebraical polynomials which involve two or more variables. 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. They are the chief instruments of research, and have themselves much benefited by being so employed. When a homogeneous polynomial is transformed by general linear substitutions as hereafter explained, and is then expressed in the original form with new coefficients affecting the new variables, certain functions of the new coefficients and variables are numerical multiples of the same functions of the original coefficients and variables. The investigation of the properties of these functions, as well for a single form as for a simultaneous set of forms, and as well for one as for many series of variables, is included in the theory of invariants. As far back as 1773 Joseph Louis Lagrange, and later Carl Friedrich Gauss, had met with simple cases of such functions, George Boole, in 1841 (Camb. Math. Journ. iii. pp. 1-20), made important steps, but it was not till 1845 that Arthur Cayley (Coll. Math. Papers, i. pp. 80-94, 95-112) showed by his calculus of hyper-determinants that an infinite series of such functions might be obtained systematically. The subject was carried on over a long series of years by himself, J. J. Sylvester, G. Salmon, L. O. Hesse, S. H. Aronhold, C. Hermite, Francesco Brioschi, R.F.A. Clebsch, P. Gordon, &c. The year 1868 saw a considerable enlargement of the field of operations. This arose from the study by Felix Klein and Sophus Lie of a new theory of groups of substitutions; it was shown that there exists an invariant theory connected with every group of linear substitutions. The invariant theory then existing was classified by them as appertaining to " finite continuous groups." Other " Galois " groups were defined whose substitution coefficients have fixed numerical values, and are particularly associated with the theory of equations. Arithmetical groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and equations, came into existence, and also particular linear and higher transformations connected with analysis and geometry. The effect of this was to co-ordinatemany branches of mathematics and greatly to increase the number of workers. The subject of transformation in general has been treated by Sophus Lie in the classical work Theorie der Transformationsgruppen. The present article is merely concerned with algebraical linear transformation. Two methods of treatment have been carried on in parallel lines, the unsymbolic and the symbolic; both of these originated with Cayley; but he with Sylvester and the English school have in the main con-fined themselves to the former, whilst Aronhold, Clebsch, Gordan, and the continental schools have principally restricted themselves to the latter. The two methods have been conducted so as to be in constant touch, though the nature of the results obtained by the one differs much from those which flow naturally from the other. Each has been- singularly successful in discovering new lines of advance and in encouraging the other to renewed efforts. P. Gordan first proved that for any system of forms there exists a finite number of covariants, in terms of which all others are expressible as rational and integral functions. This enabled David Hilbert to produce a very simple unsymbolic proof of the same theorem. - So the theory of the forms appertaining to a binary form of unrestricted order was first worked out by Cayley and P. A. MacMahon by unsymbolic methods, and later G. E. Stroh, from a knowledge of the results, was able to verify and extend the results by the symbolic method. The partition method of treating symmetrical algebra is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important theorem of expressibility is, directly we exclude unity from the partitions, a theorem concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies. The theory brought forward has not yet' found a place in any systematic treatise in any language, so that it has been judged proper to give a fairly complete account of it.' I. THE THEORY OF DETERMINANTS.' Let there be given n2 quantities all an a13 ... aln a21 a22 a23 ... a2n a31 a32 a33 .•• aan aal an2 an3... ann and form from them a product of n quantities ala a, a3y ... any, - where the first suffixes are the natural numbers 1, 2, 3, ...n taken in order, and a, )9, y, ...v is some permutation of these n numbers. This permutation by a transposition of two numbers, say a, 14, becomes $, a, y, . P, and by successively transposing pairs of letters the permutation can be reduced to the form 1, 2, 3, —II. Let k such transpositions be necessary; then the expression F'(—)kala tortsy...any, the summation being for all permutations of the n numbers, is called the determinant of the n2 quantities. The quantities ala, a21g ... are called the elements of the determinant; the term (—)kalaa,paay...anv is called a member of the determinant, and there are evidently n! members corresponding to the n! permutations of the n numbers 1, 2, 3, ... n. The determinant is usually written all an al, •.. aln a21 a22 a23 ••• a2n A= a31 an a33 ... aan and Qm2 an3 ... ann the square array being termed the matrix of the determinant. A matrix has in many parts of mathematics a signification apart from its evaluation as a determinant. A theory of matrices has been constructed by Cayley in connexion particularly with the theory of linear transformation. The matrix consists of n rows and n columns. Each row as well as each column supplies one and only one element to each member-of the determinant. Consideration of the definition of the determinant shows that the value is unaltered, when the suffixes in each element are transposed. Theorem.—If the determinant is transformed so as to read by columns as it formerly did by rows its value is unchanged. The leading member of the determinant is ana22a3s••.ann, and corresponds to the principal diagonal of the matrix. - We write frequently O = Fi t alla22a33•• •ann = (alla22a33•• •ana) • If the first two columns of the determinant be transposed the ' The elementary theory is given in the article DETERMINANT. expression for the determinant becomes E(—)kalfla2sany...anv, viz. a and (3 are transposed, and it is clear that the number of trans-positions necessary to convert the permutation $ay...v of the second suffixes to the natural order is changed by unity. Hence the trans-position of columns merely changes the sign of the determinant. Similarly it is shown that the transposition of any two columns or of any two rows merely changes the sign of the determinant. Theorem.—Interchange of any two rows or of any two columns merely changes the sign of the determinant. Corollary.—If any two rows or any two columns of a determinant be identical the value of the determinant is zero. Minors of a Determinant.—From the value of A we may separate those members which contain a particular element aik as a factor, and write the portion aik Aik; As, the cofactor of aik, is called a minor of order n — i of the determinant. Now ailAil = =aiia22ass...ann, wherein all is not to be changed, but the second suffixes in the product a22a33...a,,,, assume all per-mutations, the number of transpositions necessary determining the sign to be affixed to the member. Hence a11An =a11~ ta22a33•••ann, where the cofactor of an is clearly the determinant obtained by erasing the first row and the first column. an a33 ••• as. Hence An = a32 a33 ••• a3,, ant an3 ••• ann Similarly Aik, the cofactor of aik, is shown to be the product of (—)i+k and the determinant obtained by erasing from A the itk row and ktk column. No member of a determinant can involve more than one element from the first row. Hence we have the development A = anAii+a12Al2+a13Al3+•••+alnAln, proceeding according to the elements of the first row and the corresponding minors. Similarly we have a development proceeding according to the . elements contained in any row or in any column, viz. A =ailAil+ai2Ai2+ai3Ai3+•••+ainAin (A) ° = askAsk asxAsk+a3kA3k+... +assAnk This theory enables the evaluation of a determinant by successive reduction of the orders of the determinants involved; -1I—5 20 3I+3I0-5 =11 3 -6 I -51+3.2 -51—3.110I =3+30-30—0=3. Since the determinant a21 a22 a23 .•• a2„ an an a23 ••• a2n a31 a32 a33 ••• a3n , having two identical rows, and an2 an3 ••• ann vanishes identically; we have by development according to the elements of the first row aiiAii+a22Al2+a23A13+... +a2,Aln = 0; and, in general, since ailAil+ai2Ai2+ai3Ai3 +.•• +ainAin = A, if we suppose the itk and kih rows identical aklAi1+ak2Ai2+ak3Ai3+•••+aknAi„=0 (kCi); and proceeding by columns instead of rows, aliAlk+a2i A2k+a3i Aik+•••+an%Ank=O (kZi) identical relations always satisfied by these minors. If in the first relation of (A) we write ai.=bi.+ci.+di.+..• we find that Man Ai. = +2ci.Ai.+2di.Ai.+... so that Abreaks up into a sum of determinants, and we also obtain a theorem for the addition of determinants which have n — i rows in common. If we multiply the elements of the second row by an arbitrary magnitude a, and add to the corresponding elements of the first row, A becomes Ea,.A,.+7~Ea2.A1.=Ea1aA1.=A, showing that the value of the determinant is unchanged. In general we can prove in the same way the — Theorem.—The value of a determinant is unchanged if we add to the elements of any row or column the corresponding elements of the other rows or other columns respectively each multiplied by an arbitrary magnitude, such magnitude remaining constant in respect of the elements in a particular row or a particular column. Observation.—Every factor common to all the elements of a row or of a column is obviously a factor of the determinant, and may be taken outside the determinant brackets. Ex. gr. a2 $ 'y2 a2 — a2 y2 — a2 a f y = a $ —a y —a = I$2 a2'y2—all 1 1 1 1 0 0 fi —a y —a =(R—a)(-r—a)I$ i a 7-1a I =(s—7)(7—a)I130 y Ill _—a)(y—a)03--Os The minor Aik is ac—i— , and is itself a determinant of order n-i. We may therefore differentiate again in regard to any element ar. where r>k i,k an, a2k 2k an important expansion of A. Similarly an a2i a3i A = E aik a2k aik Al% i > k > r, air an,. an g,k. and the general theorem is manifest, and yields a. development in a sum of products of corresponding determinants. If the ja column be identical with the iik the determinant A vanishes identically; hence if j be not equal to i, k, or r, ali as; asi 0 = Z aik a2k aik Au. air as,. aar 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. Multiplication.—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,1, an, ... ann) and D = (b12, bn, bnn) may be written as a determinant of order 2n, viz. an an an ... and -1 0 0 ass ass ass ... ass 0 0 -1 an, • a*, as. ... ann 0 0 0 ... -1 ^ 0 0 ... 0 b11 biz b13... bin ^ 0 0 ... 0 b31 Inn b23 ••• b2s ^ 0 0 ... 0 b31 biz b33... b3s ^ • 0 0 ... Q bn1 tins bns ... bnn , Multiply the Yt, 2t~, ... nth rows by b11, b12, ... bin respectively, and 3 6 3 Ex. gr. 1 0 2 1 0—5 0 a12 a22 a32 an2 0 -1 0 0 0 IA B — CD for brevity. add to the (n+i)th row; by b21, b32 ••• ben, and add to the (n+2)th row; by b31, b32, ••• N. and add to the (n+3)'d row, &c. C then becomes allb11 +a12b12 +...+alnbin, a21b11 +a22b12+••• +annbin, • • . anlbll +an2b12 + ••• +annbin aub21 +a12b22 +•••+alnb2n, a21b21 +a22b22+••• +a2nbzn, •. • and b21 +an2b22 +• • • +annbin allb3l+alzb32+•••+alnb3n, a21b31+a22b32+•••+a2nb3n, ...a, ib3l+anzb32+•••+annb2n allbnl+al2bn2 +•••+alnbnn, a21bn1 +a22bn2 +•.•+aznbnn, ¢ nlbnr+¢n2bn2 +... +annbnn and all the elements of D become zero. Now by the expansion theorem the determinant becomes ()1+2+3+•••+2nB•C = ( I)n(2n+1)+' C =C. We thus obtain for the product a determinant of order n. We may say that, in the resulting determinant, the element in the ith row and kth 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 aklbil +ak2bi2 +¢k3bi3••• +¢knbtin, and we obtain other expressions by transforming either or both determinants so as to read by columns as they formerly did by rows. Remark.—In particular the square of a determinant is a determinant of the same order (bnb22b33...b,n) such that bik=bki; it is for this reason termed symmetrical. The Adjoint or Reciprocal Determinant arises from A=(a11a22a33 ..ann) by substituting for each element Aik the corresponding minor Aik so as to form D = (A11A22A33 ••• A,,,,). If we form the product i.D by the theorem for the multiplication of determinants we find that the element in the ith row and kth column of the product is arjAil+ak2Ai2+••• +aknAin, the value of which is zero when k is different from i, whilst it has the value i when k = i. Hence the product determinant has the principal diagonal elements each equal to A and the remaining elements zero. Its value is therefore A" and we have the identity D.0=A"orD=An-1. It can now be proved that the first minor of the adjoint determinant, say B„ is equal to On-2ars. From the equations rr allxi+ a12x2+ a13x3 +•••= S1 a21x1 + a22x2 + a23x 3 +••• = 2 , a31x1+a32x2+a33x3+•t• =3 , i xl = Altri + Az1tts}~~2+ Aslt:3+... , 0x2 = Al2SI + A22E2+ A32YY3+... 0x3 =A13 1+A236 +A336 +••• and thence . A"'Ei =BI1z x1+ B120x2+ Bisi x3+••• , L " 'EE2 = B21Axi+ B22,Ax2+ B23i x3+... , A"'E3 = B314xi+B320x2+B330x3+... , and comparison of the first and third systems yields Brs = An 2a,,. In general it can be proved that any minor of order p of the adjoint is equal to the complementary of the corresponding minor of the original multiplied by the (p— I)th power of the original determinant. Theorem.—The adjoint determinant is the (n—I)th power of the original determinant. The adjoint determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation. Determinants of Special Forms.—It was observed above that the square of a determinant when expressed as a determinant of the same order is such that its elements have the property expressed by aik=aki• Such determinants are called symmetrical. It is easy to see that the adjoint determinant is also symmetrical, viz. such that Aik=Aki, for the determinant got by suppressing the ith row and kth column differs only by an interchange of rows and columns from that got by suppressing the kth row and ith column. If any symmetrical determinant vanish and be bordered as shown below ¢12 X, a12 a22 a23 X2 a13 a23 a33 A3 AI A2 A3 . it is a perfect square when considered as a function of Al, A2, A3. For since A11A22—A?2=Aa33, with similar relations, we have a number of relations similar to A11A22 = A?2, and either A,.= (A,,A„) or — J (A„A,$) for all different values of r and s. Now the determinant has the value (Ai A11 +A2A22+A3A33 +2A2A3A22+2 A3A1A31 +2AiX2Al2f = —ZA;A,,—2F,A,A.Ar, in general, and hence by substitution t {A1J A11+A2 1 A22+...+X,, Ann) 2. A skew symmetric determinant has ar,=o and a,,= —as, for all values of r and s. Such a determinant when of uneven degree vanishes, for if we multiply each row by — I we multiply the determinant by (—On= -1, and the effect of this is otherwise merely to transpose the determinant so that it reads by rows as it formerlydid by columns, an operation which we know leaves the determinant unaltered. Hence A=—A or A=o. When a skew symmetric determinant is of even degree it is a perfect square. This theorem is due to Cayley, and reference may be made to Salmon's Higher Algebra, 4th ed. Art. 39. In the case of the determinant of order 4 the square root is Al2A 34 — A13A24 +A14A23• A skew determinant is one which is skew symmetric in all respects, except that the elements of the leading diagonal are not all zero. Such a determinant is of importance in the theory of orthogonal substitution. In the theory of surfaces we transform from one set of three rectangular axes to another by the substitutions )1= ax+ by+ Cz, Y = a'x+ b'y+ c'z, Z=avx+b"y+c"z, where X2+Y2+Z2=x2+y2+z2. This relation implies six equations between the coefficients, so that only three of them are independent. Further we find x =aX+a'Y+a"Z, y=bX+b'Y+b"Z, z=eX+c'Y+c"Z, and the problem is to express the nine coefficients in terms of three independent quantities. In general in space of n dimensions we have n substitutions similar to X3 =anxl+al2x2+...+alnxn, and we have to express the n2 coefficients in terms of Zn(n—1) independent quantities; which must be possible, because X+X2+...+X"2 =x1+x;l +...+4. Let there be 2n equations xi = blls1+b126+b13tt 53+•••, x2 = b21E1+b226 +b236 +••• , X1=bnlrr1+b2le2+b3,s3+•••, X2 = b1251+b22E2+b326 +•••, where b,,.= I and b,.= —b., for all values of r and s. There are then In(n—I) quantities b,,. Let the determinant of the b's be Al, and Brs, the minor corresponding to br,. We can eliminate the quantities l;i,z,•••n and obtain n relations AkX1 = (2B11—Ab)x1 +2B21x2+2B31x3+..., AkX2 = 2B12x1+(2B22—Ab)x2+2Bs2x3+..., and from these another equivalent set Abx1 = (2B11 —/b) Xl +2B12X2+2B13X3+..., Abx2 = 2B21X1+ (2B22—Ab)X2+2B23X3+..., and now writing 2Bii —Al, 2Bik Al, =aii, Ob =aik, we have a transformation which is orthogonal, because XX2=Ex2 and the elements aii, aik are functions of the Zn(n—i) independent quantities b. We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the Zn(n—I) quantities b are clearly arbitrary. For the second order we may take ~b I — A, 1 1 A2> and the adjoint determinant is the same; hence (1+X2)xi=(1—X2)X1+ 2XX2, (1+A2)x2 = -2AX1+(1 —X2)X2. Similarly, for the order 3, we take 1 v—µ 1 A =1+A2+'22+v2, µ—A 1 1+X2 v+Aµ —µ+Av - -v+Aµ 1+µ2 A+µv µ+Av—A+µv 1+0 I , leading to the orthogonal substitution 4bx1 =(1+A2—µ2—52)X1 +2(v+Aµ)X2 +2(—µ+Av)Xs Abx2 = 2{Aµ—v)XI+(1+142—A2—v2)Xz +2(µv-}•A)Xs Abx3 = 2(Av+µ)X1 +2(µv—A)X2+(1+0—A2—,22)X3. Functional determinants were first investigated by Jacobi in a work De Determinantibus Functionalibus. Suppose n dependent variables yl, y2,,..yn, each of which is a function of n independent variables xi, x2,...x,,, so that y. = f,(x,, x2,...xn). From the differential coefficients of the y's with regard to the x's we form the functional determinant we derive Al, and the adjoint is ayl a 1 ayl aY2 aY2 (Yl, Y2,...ynl R = Z—x2 ... ex — \x1, x2,...xn/ for brevity. Z.-Z. aYs ... ayn O.Z. Oxq ... If we have new variables z such that z,=¢,(yl, Y2,•••y,,), we have also z, = 4,, (x1, x2,...x,,), and we may consider the three determinants (Yl, Y21...yn) / Zl, Z2,...z„ 1 ' (sl, z2, ...z,.) x2,...xn 1\Yi, Y2r.. •yn/ 1\xi, X2, ...xn/ Forming the product of the first two by the product theorem, we obtain for the element in the it" row and kg" column azi ay1 az; ay2 azi ay,, which is a , the partial differential coefficient of z, with regard to xk. Hence the produc\/t theorem (zl, z2,...zn/ fYi, Y2,...yn' = rzi, z2,...zn) IV.. 1\ 1, Y2,...yn xlr x2r...xn xl, x2r...xn and as a particular case (yl, Y2,...yn1 el, x2,...xn = I. \Xi, x2,...xn/ \Yl, Y2,...yn Theorem.—If the functions y1, y2,•..yn be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, y1, y2,•••yn are not independent functions of xi, x2,...xn. Linear Equations.—It is of importance to study the application of the theory of determinants to the solution of a system of linear equations. Suppose given the n equations f1 =alixl+al2x2+•••+annxn =0, f2 =a2 lx1+a22x2+•.• +a2nxn =0, f s = anlxi +an2x2 +• • • +annxn = O. Denote by A the determinant (a11a22...ann). Multiplying the equations by the minors Alp., A2p,...Anp, respectively, and adding, we obtain xp (ai, Alp +a2uA2p +... +anµAnµ) = xp. = 0, since from results already giveh the remaining coefficients of x1, x2,...x I, xp+i,•••x„ vanish identically. Hence if A does not vanish xl=x2=•••=x,,=o is the only solution: but if A vanishes the equations can be satisfied by a system of values other than zeros. For in this case the n equations are not independent since identically AIpf 1+A2pf2+•••+Anpfn = 0, and assuming that the minors do not all vanish the satisfaction of n— i of the equations implies the satisfaction of the ntn Consider then the system of n — i equations a21x1+a22x2+••• +a2nxu =0 a 31x 1 +a32x2+• • • +annxn = 0 anlxl+an2x2+•••+annxn =0, which becomes on writing =y,, xx, a21y1+a22y2+•••+a2,n-lYn-l+a2n =0 aolyl+a32y2+•••+a3,n-lyn-l+a3n =0 aR1y1 +an2y2 + • •. +an,n-iyn-i+ann = O. We can solve these, assuming them 'independent, for the n—i ratios yi, Y2,•••yn-1• Now a21 An +a22Al2+• • • +a2M1Aln = 0 a31A11+a32Al2+••• +a3nAln =0 an1Al1 +an2Al2 + • • • +ann Ain = 0, and therefore, by comparison with the given equations, xi=pAli, where p is an arbitrary factor which remains constant as i varies. Hence yi = Ain where Al: and A1n are minors of the complete determinant (allan.••ann)• a21 a22 •••a2.i-1 a2,1+1 •••a2n a31 a32 •••a3,i-1 a3,i+1 ...a3n +nan1 an2 ••••aR,i-t an,i+1 ...ann Yti=(—) a21 a22 ...a2,n -1 a,1 a32 •••a3,n-II IIaRi an2 or, in words, yi is the quotient of the determinant obtained by erasing the it" column by that obtained by erasing the nt" column, multiplied by (—i)i+". For further information concerning the compatibility and independence of a system of linear equations, see Gordon, Vorlesungen fiber Invarianientheorie, Bd. i, § 8. Resultants.—When we are given k homogeneous equations in k variables or k non-homogeneous equations in k—I variables. the equations being independent, it is always possible to derive from them a single equation R=o, where in R the variables do not appear. R is a function of the coefficients which is called the " resultant " or " eliminant " of the k equations, and the process by which it is obtained is termed " elimination." We cannot combine the equations so as to eliminate the variables unless on the supposition that the equations are simultaneous, i.e. each of them satisfied by a common system of values; hence the equation R=o is derived on this supposition, and the vanishing of R expresses the condition that the equations can be satisfied by a common system of values assigned to the variables. Consider two binary equations of orders m and n respectively expressed in non-homogeneous form, viz. f(x) =f =aoxm—aix'"-l+a2xm-2—... =0, ~(x) = =box' —blx"-1+b2x"-2—... =0. If a1, a2, ...am be the roots of f=o, 01, 132, ...t3n the roots of4t=o, the condition that some root of 4 =o maypp cause f to vanish is clearly R>, =f(01)f(RR2) .f (Y..n) =0; so that Rf,o is the resultant off and 41, and expressed as a function of the roots, it is of degree m in each root 0, and of degree n in each root a, and also a symmetric function alike of the roots a and of the roots 13; hence, expressed in terms of the coefficients, it is homogeneous and of degree n in the coefficients of f, and homogeneous and of degree m in the coefficients of 0. Ex. gr. . f =aox2—a1x+a2=0, 4=box2—blx+b2. We have to multiply aol3.—aA+a2 by aA—a102+a2 and we obtain a,l P;dzaoal(R,P2+PA)+aca2A+0g)+a;$1qq$2 ala2(,+1+02)+al, qq Rp b1 qq Rq b2 p R b? -2bob2 Nl+f+2=bo, $1N2= b0, Ol+Na = - ~ b r and clearing of fractions R7,m = (aob2—a2bo)2+(albo — aobi) (aib2 —a2b1). We may equally express the result as (gal) 4,(a2)...(6(am) =0, or as This expression of R shows that, as will afterwards appear, the resultant is a simultaneous invariant of the two forms. The resultant being a product of mn root differences, is of degree mn in the roots, and hence is of weight mn in the coefficients of the forms; i.e. the sum of the suffixes in each term of the resultant is equal to mn. Resultant Expressible as a Determinant.—From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz. if aux1+a12x:+• • • +alpxv =0, a21x1 +a22x2 +• • • +629x5 = 0, ap1x1 +ap2x2 + • • • +appxp =0, be the system the condition is, in determinant form, (a11a22...app) = 0 ; in fact the determinant is the resultant of the equations. Now, suppose f and tb to have a common factor x —y, f(x) =f1(x)(x—y); 43(x) =4,i(x)(x—y), fi and ¢3 being of degrees m— r and n — i respectively ; we have the identity 01(x)f(x)=f,(x)4,(x) of degree m+n—I. Assuming then ¢1 to have the coefficients B1, B2,...Bn and fl the coefficients A1, A2,...A,n, we may equate coefficients of like powers of x in the identity, and obtain m+n homogeneous linear equations satisfied by the m+n quantities B1, B2,...Bn, Al, A2,...A,n. Forming the resultant of these equations we evidently obtain the resultant of f and 0. Thus to obtain the resultant of f =aox3+alx2+a2x+a3, 4, = box2+blx+b2 we assume the identity (Box +Bi) (aox3+a1x2+a2x+a3) = (Aox2 + Aix + A2) (box2+blx+b2), and derive the linear equations Boao —Aebo =0, Boat+Blao—A,bi—Alb° =0, Boa2 +Bl al — Aob2 — A1b1— A2bc = 0, Boa3+Bla2 — A1b2—A2b1=0, Bla3 —A2b2=0, where 624. and by elimination we obtain the resultant ao0 boo 0 a1 O b1 b0 0 a2 a1 b2 b1 bo a numerical factor as a2 0 b2 b1 being disregarded. 0aa00b2 This is Euler's method. Sylvester's leads to the same expression, but in a simpler manner. He forms n equations from f by separate multiplication by x"-1, x'J2,...x, t, in succession, and similarly treats i with m multipliers x"`-1, ,...x, 1. From these m+n equations he eliminates the m+n powers I, treating them as independent unknowns. Taking the same example as before the process leads to the system of equations acx: +aix2+a2x2+aax = 0, aoxs+alx2+asx+a' = 0, box'+bloc' +box' =0, box'+blx2 +b2x =0, bi,x2+blx+bz =0, whence by elimination the resultant ao a1 a2 as 0 0 aoa1a2a2 bo b1 b2 0 0 0 bob1b20 0 0 bo b1 b2 which reads by columns as the former determinant reads by rows, and is therefore identical with the former. E. Bezout's method gives the resultant in the form of a determinant of order m or n, according as m is n. As modified by Cayley it takes a very simple form. He forms the equation f(x)o(x') —.f(x')4(x) =0, which can be satisfied when f and tp possess a common factor. He first divides by the factor x —x', reducing it to the degree m— I in both x and x' where m>n; he then forms m equations by equating to zero the coefficients of the various powers of x'; these equations involve the m powers x°, x, of x, and regarding these as the unknowns of a system of linear equations the resultant is reached in the form of a determinant of order m. Ex. gr. Put (aox'+a1x2+a2x+aa) (box'2+bix'+b2) — (aox''+alx'2+a2x'+aa) (box'+bix+b2) =0; after division by x—x' the three equations are formed aobcx2+aobix+aob2 =0, aobix2+ (aob2 +aibi —a2bo) x+aib2 — asbo = 0, aob2x2+(alb2—asbo)x+a2b2—ask =0 and thence the resultant aobo aob1 aob2 aobi aob2 +a1b1—a2bo albs—aabo aob2 albs—aabo a2b2—asb1 which is a symmetrical determinant. Case of Three Variables.—In the next place we consider the resultants of three homogeneous polynomials in three variables. We can prove that if the three equations be satisfied by a system of values of the variable, the same system will also satisfy the Jacobian or functional determinant. For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u1 v2 WO, and by Euler's theorem of homogeneous functions xul+yu2+zua =mu xv1 +yv2 +zv3 =nv xwi+yw2+zwa = pw; denoting now the reciprocal determinant by (U1 V2 Ws) we obtain Jx=muUi+nvVi+pwWi; jy=..., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J. Further, if m=n=p, we obtain by differentiation J+xx=m \u aUl +v1+waaxl+ulUi+viVI+wiWi) . or _ (In— 1)1 +m (u+vat+waa') Hence the system of values also causes to vanish in this case; and by symmetry and also vanish. The proof being of general application we may state that a system of values which causes the vanishing of k polynomials in k variables causes also the vanishing of the Jacobian, and in particular, when the forms are of the same degree, the vanishing also of the differential coefficients of the Jacobian in regard to each of the variables. There is no difficulty in expressing the resultant by the method of symmetric functions. Taking two of the equations ax'"+(by +cz)x"'-'+... =0, a'x"+(b'y+e'z)x"—1+... = 0, we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, com- bined with the chosen values of y and z, yields a system of values which satisfies both equations. Hence in all there are mn such systems. If, therefore, we have a third equation, and we substitute each system of values in it successively and form the product of the mn expressions thus formed, we obtain a function which vanishes if any one system of values, common to the first two equations, also satisfies the third. Hence this product is the required resultant of the three equations. Now by the theory of symmetric functions, any symmetric functions of the mn values which satisfy the two equations, can be expressed in terms of the coefficient of those equations. Hence, finally, the resultant is expressed in terms of the coefficients of the three equations, and since it is at once seen to be of degree mn in the coefficient of the third equation, by symmetry it must be of degrees npand pm in the coefficients of the first and second equations respectively. Its weight will be mnp (see Salmon's Higher Algebra, 4th ed. § 77). The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner. The expression in form of a determinant presents in general considerable difficulties. If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x2, ye, z2, yz, zx, xy from the six equations u=v=w=1=1=z =0; if we apply the same process to these equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails. 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). Discriminants.—The discriminant of a homogeneous polynomial in k variables is the resultant of the k polynomials formed by differentiations in regard to each of the variables. It is the resultant of k polynomials each of degree m— l, and thus contains the coefficients of each form to the degree (m—1)k-i; hence the total degrees in the coefficients of the k forms is, by addition, k(m—1)I-'; it may further be shown that the weight of each term of the resultant is constant and equal to m(m—I)k-1 (Salmon, l.c. I00 P A binary form which has a square factor has its discriminant equal to zero. This can be seen at once because the factor in question being once repeated in both differentials, the resultant of the latter must vanish. Similarly, if a form in k variables be expressible as a quadratic function of k—I, linear functions X1, X2, ...Xr~1, the coefficients being any polynomials, it is clear that the k differentials have, in common, the system of roots derived from Xi=X2=...=Xk1=o, and have in consequence a vanishing resultant. This implies the vanishing of the discriminant of the original form. Expression in Terms of Roots.-Since xax+y, =mf, if we take any root x1, yi, of , and . substitute in mf we must obtain ; hence the resultant of and f is, disregarding Y i'i numerical factors, y1y2...Y.-1Xdiscriminant off =asXdisct. of f. Now f = (xyi —xiy) (xy2 —x2Y)... (xym—xmy), a = my'(xy2 —x2y)... (xym xmY) , and substituting in the latter any root off and forming the product, we find the resultant off and , viz. y1y2...ym(xiy2—x23,1)2(xiYs—x3yi)2...(xry, xsyr)2... and, dividing by y1Y2...ym, the discriminant of f is seen to be equal to the product of the squares of all the differences of any two roots of the equation. The discriminant of the product of two forms is equal to the product of their discriminants multiplied by the square of their resultant. This follows at' once from the fact that the discriminant is II(a aXII(1r—iie)2{II(a,.—fie)}2. II. THE THEORY OF SYMMETRIC FUNCTIONS Consider n quantities al, at, aa,...an. Every rational integral function of these quantities, which does not alter its value however the n suffixes I, 2, 3, ... n be permuted, is a rational integral symmetric function of the quantities. If we write (1 +aix) (1+ atx) ... (1 +a,,x) = +alx+aax2+...+anxn, a1, at, ...an are called the elementary symmetric functions. al = al+a2+••••+-a„ =}.•al a¢ = alas+alai+aia3+••. _ 2ala2 1—aix+a2x2—asx3+... which remains true when the symbols a and h are interchanged, as is at once evident by writing —x for x. This proves, also, that in any formula connecting al, a2, a3,... with hl, h2, ha,... the symbols a and h may be interchanged. Ex. gr, from h2=ai —a2 we derive at=hi —h2. The function Earla2P2...aPn being as above denoted by a partition of the weight, viz.(plp2...pn), it is necessary to bring under view other functions associated with the same series of numbers: such, for example, as P1 E 4' sEa1 Ps P4 Pn_2 = (pi p3)( p.p.... as ...and pn-2). The expression just written is in fact a partition of a partition, and to avoid confusion of language will be termed a separation of a partition. A partition is separated into separates so as to pro-duce a separation of the partition by writing down a set of partitions, each separate partition in its own brackets, so that when all the parts of these partitions are reassembled in a single bracket the partition which is separated is reproduced. It is convenient to write the distinct partitions or separates in descending order as regards weight. If the successive weights of the separates w1, w2, w3,••. be enclosed in a bracket we obtain a partition of the weight w which appertains to the separated partition. This partition is termed the specification of the separation. The degree of the separation is the sum of the degrees of the component separates. A separation is the symbolic representation of a product of monomial symmetric functions. A partition, (p1P1PIP3,P2P3) = (pi pPp3), can be separated in the manner (P1P2)(p1p2)(pip2)=(pip2)2(plpa , and we may take the general form of a partition to be (p1lpl2ps3...) and that of a separation (J,)il(J2)i2(J3)i3... when J1, J2, J3... denote the distinct separates involved. Theorem.— The function symbolized by (n), viz. the sum of the n'h powers of the quantities, is expressible in terms of functions which are symbolized by separations of any partition (nvinv2nv3...) 1 ! 3 of the number n. The expression is (—)v1+v2+va+...(v1+v2+i3+...—)!(n) vl.v2.v3.... _ (—\I1+'2+13+".(J1+j2+J3+...— 1)!(Jl) II(J2)12U / ji J2;J3•••. (J1)it(J2)i2(J3)i3... being a separation of (n1In"2na3...) and the summation being in regard to all such separations. For the particular case (nV Inv2nv3...) = (1n) 1 a (—)"n(n) = (—)il+is+ia+..(j1+j2+j3j+... -1)1(1)il(12)12(Is)h.... jljsa•••.To establish this write 1 +µX1+µ2X2-+3X3+... = 11 (1-1-µalxl+n2aix2+µ3aix3-I-...), the product on the right involving a factor for each, of the quantities al, at, a3..., and /c being arbitrary. Multiplying out the right-hand side and comparing coefficients Xi=(1)x1, X2= 2)x2+(12)xi, X3= 3)x3+(21)x2xi+(13)4, - X4 = (4)x4+(31)x3xl+(22)4 +(212)xzxi +(14)xi, I-'1 P2 µ3 P1 I-'2 P3 Xm =E(m3 m2 m3 ...)x,n1Y,,n2x,na .., the summation being for all partitions of m. Auxiliary Theorem.—The coefficient of xlllxi3x13... in the product 1 2 3... isI(Jl)il(J~22)a128(I.J..3)'3 ... where U1) '1(J2) '2U i3... is a separ- µ1I µ2! µ3!... j1 S) ation of (jAI1AIlha'..) of specification (mµlmµ2mµ3 1 2 3 1 2 a .••),andthesumisfor all such separations. To establish this observe the result. 1XP— "(3)'r1(21)'''2(13)'r3 v1 a2 'r2+3a3 s 7r3!7r2!2r3! x3 x2 xi and remark that (3)°1(2I)'r2(l2)'r3 is a separation of (3"12'r2I" f3'r3) of specification (3P). A similar remark may be made in respect of 1 µl 1 F12 1 P3 Xm1, A2!--'''2' µ3~Xm3' " and therefore of the product of those expressions. Hence the theorem. Now log (1+µX1+p2X2+µ2Xi+...) =E log (1+µalxi+µ2aix2+µ3aix3+...) whence, expanding by the exponential and multinomial theorems, a comparison of the2coaefficients of µn gives (n) (—) (v1+v2+v3+... -1)!xvlxv2xva vl!v2!v3!... n1 n2 n3 = E ()v1+v2+va+...-1 (v1+v2+v3+...—1)!Xn1Xv2XPs vl!v2!v3!... 1 na .. and, by the auxiliary theorem, any term X1IX,n2µzXµa ,nm3... On the right-hand side is such that the coefficient of xnixn$xng... in 1 F'1 F~2 I'3 ic1!µ2!u3!...Xn,lX,a2X,,3... IS 1 (Jl)i1(J2)72(J3)i3... ,J1J2!j3!••• where since(milm22m33...) is the specification of (J1)i1(J2)i2(J3)i3..., µl+µ2+µ3+••• =j1+j2+j3+•••• Comparison of the coefficients of xnix,2v2x; $... therefore yieids the result v1+v2+v3+... (v1+v2+v3+... -1) ! (—) v1!v2!v3!... (n) = ()il+i2+i3+..•(j1+j2+j3+...—1)!G )11 (J2)12W 3)1a..., j1;72j3•••• for the expression of Ean in terms of products of symmetric functions symbolized by separations of (n11n$2n33...). Let (n),, (n),, (n),, denote the sums of the nth powers of quantities whose elementary symmetric functions are a1, a2, a3,... ; x1, x2, x3,..; X1, X2, X3,••• respectively: then the result arrived at above from the logarithmic expansion may be written (n),.(n)x= (n)x, exhibiting (n)= as an invariant of the transformation given by the expressions of Xi, X2, X3... in terms of x1, x3, x4,.... The inverse question is the expression of any monomial symmetric function by means of the power functions (r) =s,. Theorem of Reciprocity: If X,,,1Xm2Xma... =...+8(sl Is2 2s3 3...)x 1Ix12x=3...-{-..., where 0 is a numerical coefficient, then also of 02 v3 P1 P3 P2 Al A2 hi X,1 X,2 X,3. ..+e(m1 mz ma ...)x11 xy2 X13 ...+.... We have found above that the coefficient of (x111 x122x,3...) in the product XmiX,µn~Xn",;... is µ1!µ2!µ3!... Y (J1)il(J2)iz(J3)i.... j1 lj2ij3!... the sum being for all separations of 1h11l2h-lhaa...) which have the specification (mi1ml2`zmg3...). We can multiply out this expression so as to obtain a series of monomials of the form 0(s11sN'3...). It can be shown that the number B enumerates distributions of a certain nature defined by the partitions (milms2...), (sils~z ), an = alai a3... aa. The general monomial symmetric function is EaP1P2aP3...ann, the summation being for all permutations of the indices which result in different terms. The function is written (Plp2p3•••pn) for brevity, and repetitions of numbers in the bracket are indicated by exponents, so that (PIPi z) is written (pips). The weight of the function is the sum of the numbers in the bracket, and the degree , the highest of those numbers. Ex. gr. The elementary functions are denoted by (1), 12 (12), 1n , are all of the first degree, nd are of eights 1, 2, 3,...n respectively. Remark.—In this notation (0)=tai=(1;(02)=144 =(2'');...(0•) = (; ), &c. The binomial coefficients appear, in fact, as symmetric functions, and this is frequently of importance. The order of the numbers in the bracket (¢lps...pn) is immaterial; we may therefore always place them, as is most convenient, in descending order of magnitude; the numbers then constitute an ordered partition of the weight w, and the leading number denotes the degree. The sum of the monomial functions of a given weight is called the homogeneous-product-sum or complete symmetric function of that weight; it is denoted by h.; it is connected with the elementary functions by the formula 1 =1-{-hlx+hsx2+hax3+..., (li'l =...) and it is seen intuitively that the number 0 remains unaltered when the first two of these partitions are interchanged (see COMBINATORIAL ANALYSIS). Hence the theorem is established. Putting xi =1 and x2=x3=x4=...=o, we finl a particular law of reciprocity given by Cayley and Betti, (1m')gl(1'm2)M2(1"'a)µa,•. =...+o(si's2c 2s83...)+•.., (1'1) °1(1 a2) ca(1,3) °a... =... 0(mi'm2 zm$ 3...) +... ; and another by putting x1= x2 = x3 =... =1, for then Xm becomes hm, and we have hih,',3hme... =... +o' (si's= ssa 3...) +..., o' m"`lynµ2ru" 3... Theorem of Expressibility.—" If a symmetric function be symboilized by (Xµv...) and (X2X2X3...), (µ1112113...), (vlv2v3...)... be any partitions of a, µ, v... respectively, the function (XAs ...) is expressible by means of functions symbolized by separation of ()\1X2X3...µiµzµ3... vlv2v3...)." For, writing as before, =}.FO(S°'S°3S°3...)x~'x12x~3 "'1 m2 m3' 1 2 8 1 S $= EPxi 1x1 sxAa P is a linear function of separations of (1'11'2123...) of specification (ml"'m2 2ms3...), and if = EP'x!'x13xi 3..., P' is a linear 1 3 31 S a function of separations of (l21122le3...) of specification (sl's=2sis...). Suppose the separations of (li 1122183...) to involve k different specifications and form the k identities 1aX,'2eX'n3a"`3a... =EP(a)xatxa2x"3 (s=1> 2, •..k) le n'2e I e2 ,3... f where (mi''m2 Rams.'•..) is one of the k specifications. The law of reciprocity shows that t-s P(e)=161at(mµltmP2tre8t ") It Et $t ' a. e~1 viz.: a linear function of symmetric functions symbolized by the k specifications ; and that oa, = Bo. A table may be formed expressing the k expressions Po), P(2),...P(k) as linear functions of the k expressions (m1µ"mµ2a a 2am"+sa 3e...), s =1, 2, ...k, and the numbers oa, occurring therein possess row and column symmetry. By solving k linear equations we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical. Theorem.-" The symmetric function (mi 18m2 tams 3e .••) whose partition is a specification of a separation of the function symbolized by (li'132133...) is expressible as a linear function of symmetric functions symbolized by separations of (li'122133...) and a symmetrical table may be thus formed." It is now to be remarked that the partition (lA'1A21A3•..) can be derived from 1 2 3 le 2a ae by substituting for the numbers m1e, m2,, m3e,... certain partitions of those numbers (vide the definition of the specification of a separation). Hence the theorem of expressibility enunciated above. A new statement of the law of reciprocity can be arrived at as follows: Since. P(') =µ11!µ2.!µ8al... (J1)'I(J2)12(J3)is- where (J1)I1(J2)t2(Ja)13... is a separation of (1 132133...) of specification (mµ'"mµ2amµ3a...), placing s under the summation sign to denote the le Ya Sa specification involved, µ1.!µ2,!µg8..• (J1)'1j(J2)/2(J3)7a... l!js!js!•• e—1 µlt!/a2t!/1s,!... (Pit (JsY2 (J 3)13...— Bts(mµl.mµsamµs....), j1;~2;~$••• 1a Ya $a where o,t =913. ° °-1 Theorem of Symmetry.—If we form the separation function (J1)''(Js)'2(J8)1a..• jl!j2!i3!••• appertaining to the function (li'122le3...), each separation having a specification (n1"`1"mt,2'm"`ae31..•)' multiply by /111! A,! z µs and take therein the coefficient of the function (m su tml' tm s3 t...), we obtain the same result as if we formed the separation function in regard to the specification m"'tmµ2tmµ3, multiplied b ( 1t ,z a, ...), y µ1t. µ2t• µat•••• and took therein the coefficient of the function (m"le"m2e'21m'a,'3e...). Ex.gr., take (lAll ...)=(214);(mi;ams,'...) = (321);(+nii trnsi t...)=(318) ; we find (21)(12)(1)+(13)(2)(l) =...+13(313)+..., (21)(1)3=...+13(321)+...The Differential Operators.—Starting with the relation (1 +aix) (1 +a2x)... (1 +a»x) = 1 +alx +82x2 +... +a,.x" multiply each side by I +tax, thus introducing a new quantity µ; we obtain (1+ age) (1-+2x)...(1+a,.x) (1+µx) =1+(a1+µ)x+(a2+pal) x2+.•• so that Ash, a2, a3,...a1) = f, a rational integral function of the ele- mentary functions, is converted into f(a1+a2+µ(11,...a„+µ¢n_1) =f +I J1f+~;alf+Td f+... where a a a a d1=aa1+81882+82883+...+a„_la¢n and di denotes, not s successive operations of dl, but the operator of order s obtained by raising dl to the so' power symbolically as in Taylor's theorem in the Differential Calculus. Write also ;id, = D, so that f(a1+µ. as +µa1, ...¢,.+P¢,.-1) =f+/Dlf+la2D2f+I;3Daf+.... The introduction of the quantity µ converts the symmetric function (81X282...) into (X1Xsas+...) +µA' (~s71a...) +AA3(a1X3...) +/Aa(A1a2...) +.... Hence, if f(a1, as, ...a„) = (XiA2X$...) +/A1(A2X3... J +1412(XlX3...) +ta'A3(X1X2...) +... (1+µD1+µ2 2+µ3D3+...)(XIX2X3•..)• Comparing coefficients of like powers of p we obtain
End of Article: ALGEBRAIC FORMS
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