220 ALGEBRA 5.7.4-2. Extremal properties of quadratic forms. A point x 0 on a smooth surface S is called a stationary point of a differentiable function f defined on S if the derivative of f at the point x 0 in any direction on S is equal to zero. The value f (x 0 ) of the function f at a stationary point x 0 is called its stationary value. The unit sphere in a Euclidean space V is the set of all x V such that x ⋅ x = 1 (x = 1). (5.7.4.1) T HEOREM. Let B(x, x) be a real quadratic form and let B(x, y)=(Ax) ⋅ y be the corresponding polar bilinear form, where A is a Hermitian operator. The stationary values of the quadratic form B(x, x) on the unit sphere (5.7.4.1) coincide with eigenvalues of the operator A . These stationary values are attained, in particular, on the unit eigenvectors e k of the operator A . Remark. If the eigenvalues of the operator A satisfy the inequalities λ 1 ≥ ≥ λ n ,thenλ 1 and λ n are the largest and the smallest values of B(x, x) on the sphere x ⋅ x = 1. 5.7.5. Second-Order Hypersurfaces 5.7.5-1. Definition of a second-order hypersurface. A second-order hypersurface in an n-dimensional Euclidean space V is the set of all points x V satisfying an equation of the form A(x, x)+2B(x)+c = 0,(5.7.5.1) where A(x, x) is a real quadratic form different from identical zero, B(x) is a linear form, and c is a real constant. Equation (5.7.5.1) is called the general equation of a second-order hypersurface. Suppose that in some orthonormal basis i 1 , , i n ,wehave A(x, x)=X T AX = n  i,j=1 a ij x i x k , B(x)=BX = n  i=1 b i x i , X T =(x 1 , , x n ), A ≡ [a ij ], B =(b 1 , , b n ). Then the general equation (5.7.5.1) of a second-order hypersurface in the Euclidean space V with the given orthonormal basis i 1 , , i n can be written as X T AX + 2BX + c = 0. The term A(x, x)=X T AX is called the group of the leading terms of equation (5.7.5.1), and the terms B(x)+c = BX + c are called the linear part of the equation. 5.7.5-2. Parallel translation. A parallel translation in a Euclidean space V is a transformation defined by the formulas X = X  + ◦ X,(5.7.5.2) where ◦ X is a fixed point, called the new origin. 5.7. BILINEAR AND QUADRATIC FORMS 221 In terms of coordinates, (5.7.5.2) takes the form x k = x  k + ◦ x k (k = 1, 2, , n), where X T =(x 1 , , x n ), X T =(x  1 , , x  n ), ◦ X T =( ◦ x 1 , , ◦ x n ). Under parallel translations any basis remains unchanged. The transformation of the space V defined by (5.7.5.2) reduces the hypersurface equation (5.7.5.1) to A(x  , x  )+2B  (x  )+c  = 0, where the linear form B  (x  ) and the constant c  are defined by B  (x  )=A(x  , ◦ x )+B(x  ), c  = A( ◦ x , ◦ x )+2B( ◦ x )+c, or, in coordinate notation, B  (x  ) ≡ B  X  = n  i=1 b  i x  i , c  = n  i=1 (b  i + b i ) ◦ x i + c, b  i = n  j=1 a ij ◦ x j + b i . Under parallel translation the group of the leading terms preserves its form. 5.7.5-3. Transformation of one orthonormal basis into another. The transition from one orthonormal basis i 1 , , i n to another orthonormal basis i  1 , , i  n is defined by an orthogonal matrix P ≡ [p ij ]ofsizen × n, i.e., i  i = n  j=1 p ij i j (i = 1, 2, , n). Under this orthogonal transformation, the coordinates of points are transformed as follows: X  = PX, or, in coordinate notation, x  k = n  i=1 p ki x i (k = 1, 2, , n), (5.7.5.3) where X T =(x 1 , , x n ), X T =(x  1 , , x  n ). If the transition from the orthonormal basis i 1 , , i n to the orthonormal basis i  1 , , i  n is defined by an orthogonal matrix P , then the hypersurface equation (5.7.5.1) in the new basis takes the form A  (x  , x  )+2B  (x  )+c  = 0. The matrix A  ≡ [a  ij ](A  (x  , x  )=X T A  X  ) is found from the relation A  = P –1 AP . Thus, when passing from one orthonormal basis to another orthonormal basis, the matrix of a quadratic form is transformed similarly to the matrix of some linear operator. Note that the operator A whose matrix in an orthonormal basis coincides with the matrix of the quadratic form A(x, x) is Hermitian. 222 ALGEBRA The coefficients b  i of the linear form B  (x  )= n  i=1 b  i x  i are found from the relations [to this end, one should use (5.7.5.3)] n  i=1 b  i x  i = n  i=1 b i x i , and the constant is c  = c. 5.7.5-4. Invariants of the general equation of a second-order hypersurface. An invariant of the general second-order hypersurface equation (5.7.5.1) with respect to parallel translations and orthogonal transformations of an orthogonal basis is, by definition, any function f(a 11 , , a nn , b 1 , , b n , c)ofthecoefficients of this equation that does not change under such transformations of the space. T HEOREM. The coefficients of the characteristic polynomial of the matrix A of the quadratic form A(x, x) and the determinant det  A of the block matrix  A =  AB B T c  are invariants of the general second-order hypersurface equation (5.7.5.1). Remark. The quantities det A,Tr(A), rank (A), and rank (  A) are invariants of equation (5.7.5.1). 5.7.5-5. Center of a second-order hypersurface. The center of a second-order hypersurface is a point ◦ x such that the linear form B  (x  ) becomes identically equal to zero after the parallel translation that makes ◦ x the new origin. Thus, the coordinates of the center can be found from the system of equations of the center of a second-order hypersurface n  j=1 a ij ◦ x j + b i = 0 (i = 1, 2, , n). If the center equations for a hypersurface S have a unique solution, then S is called a central hypersurface. If a hypersurface S has a center, then S consists of pairs of points, each pair being symmetric with respect to the center. Remark 1. For a second-order hypersurface S with a center, the invariants det A,det  A, and the free term c  are related by det  A = c  det A. Remark 2. If the origin is shifted to the center of a central hypersurface S, then the equation of that hypersurface in new coordinates has the form A(x, x)+ det  A det A = 0. 5.7. BILINEAR AND QUADRATIC FORMS 223 5.7.5-6. Simplification of a second-order hypersurface equation. Let A be the operator whose matrix in an orthonormal basis i 1 , , i n coincides with the matrix of a quadratic form A(x, x). Suppose that the transition from the orthonormal basis i 1 , , i n to the orthonormal basis i  1 , , i  n is defined by an orthogonal matrix P , and A  = P –1 AP is a diagonal matrix with the eigenvalues of the operator A on the main diagonal. Then the equation of the hypersurface (5.7.5.1) in the new basis takes the form n  i=1 λ i x  i 2 + 2 n  i=1 b  i x  i + c = 0,(5.7.5.4) where the coefficients b  i are determined by the relations n  i=1 b  i x  i = n  i=1 b i x i . The reduction of any equation of a second-order hypersurface S to the form (5.7.5.4) is called the standard simplification of this equation (by an orthogonal transformation of the basis). 5.7.5-7. Classification of central second-order hypersurfaces. 1 ◦ .Leti 2 , , i n be an orthonormal basis in which a second-order central hypersurface is defined by the equation (called its canonical equation) n  i=1 ε i x 2 i a 2 i +sign det  A det A = 0,(5.7.5.5) where x 1 , , x n are the coordinates of x in that basis, and the coefficients ε 1 , , ε n take the values –1, 0,or1. The constants a k > 0 are called the semiaxes of the hypersurface. The equation of any central hypersurface S can be reduced to the canonical equation (5.7.5.5) by the following transformations: 1. By the parallel translation that shifts the origin to the center of the hypersurface, its equation is transformed to (see Paragraph 5.7.5-5): A(x, x)+ det  A det A = 0. 2. By the standard simplification of the last equation, one obtains an equation of the hypersurface in the form n  i=1 λ i x 2 i + det  A det A = 0. 3. Letting 1 a 2 k = ⎧ ⎨ ⎩ |λ k | | det A| | det  A| if det  A ≠ 0, |λ k | if det  A = 0, ε k =signλ k (k = 1, 2, , n), one passes to the canonical equation (5.7.5.5) of the central second-order hypersurface. 224 ALGEBRA 2 ◦ .Letp be the number of positive eigenvalues of the matrix A and q the number of negative ones. Central second-order hypersurfaces admit the following classification. A hypersurface is called: a)an(n–1)-dimensional ellipsoid if p = n and sign det  A det A =–1,orq = n and sign det  A det A = 1 ∗ ; b)animaginary ellipsoid if p = n and sign det  A det A = 1,orq = n and sign det  A det A =–1; c)ahyperboloid if 0 < p < n (0 < q < n) and sign det  A det A ≠ 0; d) degenerate if sign det  A det A = 0. 5.7.5-8. Classification of noncentral second-order hypersurfaces. 1 ◦ .Leti 2 , , i n be an orthonormal basis in which a noncentral second-order hypersurface is defined by the equation (called its canonical equation) p  i=1 λ i x 2 i + 2μx n + c  = 0,(5.7.5.6) where x 1 , , x n are the coordinates of x in that basis: p =rank(A). The equation of any noncentral second-order hypersurface S can be reduced to the canonical form (5.7.5.6) by the following transformations: 1. If p =rank(A), then after the standard simplification and renumbering the basis vectors, equation (5.7.5.1) turns into p  i=1 λ i x  i 2 + 2 p  i=1 b  i x  i + 2 n  i=p+1 b  i x  i + c = 0. 2. After the parallel translation x  k = ⎧ ⎨ ⎩ x  k + b  k λ k for k = 1, 2, , p, x  k for k = p + 1, p + 2, , n, the last equation can be represented in the form p  i=1 λ i x  i 2 + 2 n  i=p+1 b  i x  i + c  = 0, c  = c – p  i=1 b  i λ i . 3. Leaving intact the first p basis vectors and transforming the last basis vectors so that the term n  i=p+1 b  i x  i turns into μx  n , one reduces the hypersurface equation to the canonical form (5.7.5.6). 2 ◦ . Noncentral second-order hypersurfaces admit the following classification. A hypersurface is called: a)aparaboloid if μ ≠ 0 and p = n – 1; in this case, the parallel translation in the direction of the x n -axis by – c  2μ yields the canonical equation of a paraboloid n–1  i=1 λ i x 2 i + 2μx n = 0; ∗ If a 1 = = a n = R, then the hypersurface is a sphere of radius R in n-dimensional space. 5.8. SOME FACTS FROM GROUP THEORY 225 b)acentral cylinder if μ = 0, p < n; its canonical equation has the form p  i=1 λ i x 2 i + c  = 0; c)aparaboloidal cylinder if μ ≠ 0, p < n – 1; in this case, the parallel translation along the x n -axis by – c  2μ yields the canonical equation of a paraboloidal cylinder p  i=1 λ i x 2 i + 2μx n = 0. 5.8. Some Facts from Group Theory 5.8.1. Groups and Their Basic Properties 5.8.1-1. Composition laws. Let T be a mapping defined on ordered pairs a, b of elements of a set A and mapping each pair a, b to an element c of A. In this case, one says that a composition law is definedonthe set A. The element c A is called the composition of the elements a, b A and is denoted by c = aTb. A composition law is commonly expressed in one of the two forms: 1. Additive form: c = a + b; the corresponding composition law is called addition and c is called the sum of a and b. 2. Multiplicative form: c = ab; the corresponding composition law is called multiplication and c is called the product of a and b. A composition law is said to be associative if aT(bTc)=(aTb)Tc for all a, b, c A. In additive form, this relation reads a +(b + c)=(a + b)+c; and in multiplicative form, a(bc)=(ab)c. A composition law is said to be commutative if aTb = bTa for all a, b A. In additive form, this relation reads a + b = b + a; and in multiplicative form, ab = ba. An element e of the set A is said to be neutral with respect to the composition law T if aTe = a for any a A. In the additive case, a neutral element is called a zero element, and in the multiplicative case, an identity element. An element b is called an inverse of a A if aTb = e. The inverse element is denoted by b = a –1 . In the additive case, the inverse element of a is called the negative of a and it is denoted by –a. Example 1. Addition and multiplication of real numbers are composition laws on the set of real numbers. Both these laws are commutative. The neutral element for the addition is zero. The neutral element for the multiplication is unity. 226 ALGEBRA 5.8.1-2. Notion of a group. A group is a set G with a composition law T satisfying the conditions: 1. The law T is associative. 2. There is a neutral element e G. 3. For any a G, there is an inverse element a –1 . A group G is said to be commutative or abelian if its composition law T is commutative. Example 2. The set Z of all integer numbers is an abelian group with respect to addition. The set of all positive real numbers is an abelian group with respect to multiplication. Any linear space is an abelian group with respect to the addition of its elements. Example 3. Permutation groups.LetE be a set consisting of finitely many elements a, b, c, A permutation of E is a one-to-one mapping of E onto itself. A permutation f of the set E can be expressed in the form f =  a b c f(a) f(b) f(c)  . On the set P of all permutations of E, the composition law is introduced as follows: if f 1 and f 2 are two permutations of E, then their composition f 2 ◦ f 1 is the permutation obtained by consecutive application of f 1 and f 2 . This composition law is associative. The set of all permutations of E with this composition law is a group. Example 4. The group Z 2 that consists of two elements 0 and 1 with the multiplication defined by 0 ⋅ 0 = 0, 0 ⋅ 1 = 1, 1 ⋅ 0 = 1, 1 ⋅ 1 = 1 and the neutral element 0 is called the group of modulo 2 residues. Properties of groups: 1. If aTa –1 = e,thena –1 Ta = e. 2. eTa = a for any a. 3. If aTx = e and aTy = e,thenx = y. 4. The neutral element e is unique. 5.8.1-3. Homomorphisms and isomorphisms. Recall that a mapping f : A → B of a set A into a set B is a correspondence that associates each element of A with an element of B.Therange of the mapping f is the set of all b B such that b = f (a). One says that f is a one-to-one mapping if it maps different elements of A into different elements of B, i.e., for any a 1 , a 2 A such that a 1 ≠ a 2 ,we have f (a 1 ) ≠ f(a 2 ). A mapping f : A → B is called a mapping of the set A onto the set B if each element of B is an image of some element of A, i.e., for any b B,thereisa A such that b = f(a). A mapping f of A onto B is said to be invertible if there is a mapping g : B → A such that g(f (a)) = a for any a A. The mapping g is called the inverse of the mapping f and is denoted by g = f –1 . For definiteness, we use the multiplicative notation for composition laws in what follows, unless indicated otherwise. Let G be a group and let  G be a set with a composition law. A mapping f : G →  G is called a homomorphism if f(ab)=f(a)f(b)foralla, b G; and the subset of  G consisting of all elements of the form f (a), a G, is called a homo- morphic image of the group G and is denoted by f(G). Note that here the set  G with a composition law is not necessarily a group. However, the following result holds. . space. T HEOREM. The coefficients of the characteristic polynomial of the matrix A of the quadratic form A(x, x) and the determinant det  A of the block matrix  A =  AB B T c  are invariants of the general. relations n  i=1 b  i x  i = n  i=1 b i x i . The reduction of any equation of a second-order hypersurface S to the form (5.7.5.4) is called the standard simplification of this equation (by an orthogonal transformation of the basis). 5.7.5-7 group and let  G be a set with a composition law. A mapping f : G →  G is called a homomorphism if f(ab)=f(a)f(b)foralla, b G; and the subset of  G consisting of all elements of the form f