5.7. BILINEAR AND QUADRATIC FORMS 213 5.6.3-6. Canonical form of linear operators. An element x is calledan associated vector of an operator Acorresponding to its eigenvalue λ if for some m ≥ 1,wehave (A – λI) m x ≠ 0,(A – λI) m+1 x = 0. The number m is called the order of the associated vector x. T HEOREM. Let A be a linear operator in an n -dimensional unitary space V . Then there is a basis {i m k } ( k = 1, 2, , l , m = 1, 2, , n k , n 1 + n 2 + ···+ n l = n )in V consisting of eigenvectors and associated vectors of the operator A such that the action of the operator A is determined by the relations Ai 1 k = λ k i 1 k (k = 1, 2, , l), Ai m k = λ k i m k + i m–1 k (k = 1, 2, , l, m = 2, 3, , n k ). Remark 1. The vectors i 1 k (k = 1, 2, , l) are eigenvectors of the operator A corresponding to the eigenvalues λ k . Remark 2. The matrix A of the linear operator A in the basis {i m k } has canonical Jordan form, and the above theorem is also called the theorem on the reduction of a matrix to canonical Jordan form. 5.7. Bilinear and Quadratic Forms 5.7.1. Linear and Sesquilinear Forms 5.7.1-1. Linear forms in a unitary space. A linear form or linear functional on V is a linear operator A in L(V, C), where C is the complex plane. T HEOREM. For any linear form f in a finite-dimensional unitary space V ,thereisa unique element h in V such that f(x)=x ⋅ h for all x V. Remark. This statement is true also for a Euclidean space V and a real-valued linear functional. 5.7.1-2. Sesquilinear forms in unitary space. A sesquilinear form on a unitary space V is a complex-valued function B(x, y)oftwo arguments x, y V such that for any x, y, z in V and any complex scalar λ, the following relations hold: 1. B(x + y, z)=B(x, z)+B(y, z). 2. B(x, y + z)=B(x, y)+B(x, z). 3. B(λx, y)=λB(x, y). 4. B(x, λy)= ¯ λB(x, y). 214 ALGEBRA Remark. Thus, B(x, y) is a scalar function that is linear with respect to its first argument and antilinear with respect to its second argument. For a real space V, sesquilinear forms turn into bilinear forms (see Paragraph 5.7.2). THEOREM. Let B(x, y) be a sesquilinear form in a unitary space V . Then there is a unique linear operator A in L(V, V) such that B(x, y)=x ⋅ (Ay). C OROLLARY. If B(x, y) is a sesquilinear form in a unitary space V , then there is a unique linear operator A in L(V, V) such that B(x, y)=(Ax) ⋅ y. 5.7.1-3. Matrix of a sesquilinear form. Any sesquilinear form B(x, y)onann-dimensional linear space with a given basis e 1 , , e n can be uniquely represented as B(x, y)= n i,j=1 b ij ξ i ¯η j , b ij = B(e i , e j ), and ξ i , η j are the coordinates of x and y in the given basis. The matrix B ≡ [b ij ]ofsize n × n is called the matrix of the sesquilinear form B(x, y) in the given basis e 1 , , e n . This sesquilinear form can also be represented as B(x, y)=X T BY , X T ≡ (ξ 1 , , ξ n ), Y T ≡ (¯η 1 , , ¯η n ). 5.7.2. Bilinear Forms 5.7.2-1. Definition of a bilinear form. A bilinear form on a real linear space V is a real-valued function B(x, y)oftwoarguments x L, y V satisfying the following conditions for any vectors x, y,andz in V and any real λ: 1. B(x + y, z)=B(x, z)+B(y, z). 2. B(x, y + z)=B(x, y)+B(x, z). 3. B(λx, y)=B(x, λy)=λB(x, y). T HEOREM. Let B(x, y) be a bilinear form in a Euclidean space V . Then there is a unique linear operator A in L(V, V) such that B(x, y)=(Ax) ⋅ y. A bilinear form B(x, y)issaidtobesymmetric if for any x and y,wehave B(x, y)=B(y, x). A bilinear form B(x, y)issaidtobeskew-symmetric if for any x and y,wehave B(x, y)=–B(y, x). Any bilinear form can be represented as the sum of symmetric and skew-symmetric bilinear forms. T HEOREM. A bilinear form B(x, y) on a Euclidean space V is symmetric if and only if the linear operator A in the representation (5.6.6.1) is Hermitian ( A = A ∗ ). 5.7. BILINEAR AND QUADRATIC FORMS 215 5.7.2-2. Bilinear forms in finite-dimensional spaces. Any bilinear form B(x, y)onann-dimensional linear space with a given basis e 1 , , e n can be uniquely represented as B(x, y)= n i,j=1 b ij ξ i η j , b ij = B(e i , e j ), and ξ i , η j are the coordinates of the vectors x and y in the given basis. The matrix B ≡ [b ij ] of size n × n is called the matrix of the bilinear form in the given basis e 1 , , e n .The bilinear form can also be represented as B(x, y)=X T BY , X T ≡ (ξ 1 , , ξ n ), Y T ≡ (η 1 , , η n ). Remark. Any square matrix B ≡ [b ij ] can be regarded as a matrix of some bilinear form in a given basis e 1 , , e n . If this matrixis symmetric(skew-symmetric), then the bilinear form issymmetric (skew-symmetric). The rank of a bilinear form B(x, y)onafinite-dimensional linear space L is defined as the rank of the matrix B of this form in any basis: rank B(x, y)=rank(B). A bilinear form on a finite dimensional space V is said to be nondegenerate (degenerate) if its rank is equal to (is less than) the dimension of the space V, i.e., rank B(x, y)=dimV (rank B(x, y)<dimV). 5.7.2-3. Transformation of the matrix of a bilinear form in another basis. Suppose that the transition from a basis e 1 , , e n to a basis e 1 , , e n is determined by the matrix U ≡ [u ij ]ofsizen × n,i.e. e i = n j=1 u ij e j (i = 1, 2, , n). T HEOREM. The matrices B and B of a bilinear form B(x, y) in the bases e 1 , , e n and e 1 , , e n , respectively, are related by B = U T BU. 5.7.2-4. Multilinear forms. A multilinear form on a linear space V is a scalar function B(x 1 , , x p )ofp arguments x 1 , , x p V, which is linear in each argument for fixed values of the other arguments. A multilinear form B(x, y)issaidtobesymmetric if for any two arguments x l and x l , we have B(x 1 , , x k , , x l , , x p )=B(x 1 , , x l , , x k , , x p ). A multilinear form B(x, y)issaidtobeskew-symmetric if for any two arguments x l and x l , we have B(x 1 , , x k , , x l , , x p )=–B(x 1 , , x l , , x k , , x p ). 216 ALGEBRA 5.7.3. Quadratic Forms 5.7.3-1. Definition of a quadratic form. A quadratic form on a real linear space is a scalar function B(x, x) obtained from a bilinear form B(x, y)forx = y. Any symmetric bilinear form B(x, y)ispolar with respect to the quadratic form B(x, x). These forms are related by B(x, y)= 1 2 [B(x + y, x + y)–B(x, x)–B(y, y)]. 5.7.3-2. Quadratic forms in a finite-dimensional linear space. Any quadratic form B(x, x)inann-dimensional linear space with a given basis e 1 , , e n can be uniquely represented in the form B(x, x)= n i,j=1 b ij ξ i ξ j ,(5.7.3.1) where ξ i are the coordinates of the vector x in the given basis, and B ≡ [b ij ] is a symmetric matrix of size n × n, called the matrix of the bilinear form B(x, x) in the given basis. This quadratic form can also be represented as B(x, x)=X T BX, X T ≡ (ξ 1 , , ξ n ). Remark. Any quadratic form can be represented in the form (5.7.3.1) with infinitely many matrices B such that B(x, x)=X T BX. In what follows, we consider only one of such matrices, namely, the symmetric matrix. A quadratic form is real-valued if its symmetric matrix is real. A real-valued quadratic form B(x, x) is said to be: a) positive definite (negative definite)ifB(x, x)>0 (B(x, x)<0)foranyx ≠ 0; b) alternating if there exist x and y such that B(x, x)>0 and B(y, y)<0; c) nonnegative (nonpositive)ifB(x, x) ≥ 0 (B(x, x) ≤ 0)forallx. If B(x, y) is a polar bilinear form with respect to some positive definite quadratic form B(x, x ), then B(x, y) satisfies all axioms of the scalar product in a Euclidean space. Remark. The axioms of the scalar product can be regarded as the conditions that determine a bilinear form that is polar to some positive definite quadratic form. The rank of a quadratic form on a finite-dimensional linear space V is, by definition, the rank of the matrix of that form in any basis of V,rankB(x, x)=rank(B). A quadratic form on a finite-dimensional linear space V is said to be nondegenerate (degenerate) if its rank is equal to (is less than) the dimension of V, i.e., rank B(x, x)=dimV (rank B(x, x)<dimV). 5.7.3-3. Transformation of a bilinear form in another basis. Suppose that the transition from the basis e 1 , , e n to the basis e 1 , , e n is given by the matrix U ≡ [u ij ]ofsizen × n,i.e. e i = n j=1 u ij e j (i = 1, 2, , n). Then the matrices B and B of the quadratic form B(x, x) in the bases e 1 , , e n and e 1 , , e n , respectively, are related by B = U T BU. 5.7. BILINEAR AND QUADRATIC FORMS 217 5.7.3-4. Canonical representation of a real quadratic form. Let g 1 , , g n be a basis in which the real quadratic form B(x, x) in a linear space V admits the representation B(x, x)= n i=1 λ i η 2 i ,(5.7.3.2) where η 1 , , η n are the coordinates of x in that basis. This representation is called a canonical representation of the quadratic form, the real coefficients λ 1 , , λ n are called the canonical coefficients, and the basis g 1 , , g n is called the canonical basis. The number of nonzero canonical coefficients is equal to the rank of the quadratic form. T HEOREM. Any real quadratic form on an n -dimensional real linear space V admits a canonical representation (5.7.3.2). 1 ◦ . Lagrange method. The basic idea of the method consists of consecutive transformations of the quadratic form: on every step, one should single out the perfect square of some linear form. Consider a quadratic form B(x, x)= n i,j=1 b ij ξ i ξ j . Case 1. Suppose that for some m (1 ≤ m ≤ n), we have b mm ≠ 0. Then, letting B(x, x)= 1 b mm n k=1 b mk ξ k 2 + B 2 (x, x), one can easily verify that the quadratic form B 2 (x, x) does not contain the variable ξ m . This method of separating a perfect square in a quadratic form can always be applied if the matrix [b ij ](i, j = 1, 2, , n) contains nonzero diagonal elements. Case 2. Suppose that b mm = 0, b ss = 0,butb ms ≠ 0. In this case, the quadratic form can be represented as B(x, x)= 1 2b sm n k=1 (b mk + b sk )ξ k 2 – 1 2b sm n k=1 (b mk – b sk )ξ k 2 + B 2 (x, x), where B 2 (x, x) does not contain the variables ξ m , ξ s , and the linear forms in square brackets are linearly independent (and therefore can be taken as new independent variables or coordinates). By consecutive combination of the above two procedures, the quadratic form B(x, x) can always be represented in terms of squared linear forms; these forms are linearly independent, since each contains a variable which is absent in the other linear forms. 2 ◦ . Jacobi method. Suppose that Δ 1 ≡ b 11 ≠ 0, Δ 2 ≡ b 11 b 12 b 21 b 22 ≠ 0, , Δ n ≡ det B ≠ 0, where B ≡ [b ij ] is the matrix of the quadratic form B(x, x)insomebasise 1 , , e n .One can obtain a canonical representation of this form using the formulas λ 1 = Δ 1 , λ i = Δ i Δ i–1 (i = 2, 3, , n). 218 ALGEBRA The basis e 1 , , e n is transformed to the canonical basis g 1 , , g n by the formulas g i = n j=1 α ij e j (i = 1, 2, , n), α ij =(–1) i+j Δ i–1,j Δ i–1 , where Δ i–1,j is the minor of the submatrix of B ≡ [b ij ] formed by the elements on the intersection of its rows with indices 1, 2, , i – 1 and columns with indices 1, 2, , j – 1, j + 1, i. 5.7.3-5. Normal representation of a real quadratic form. Let g 1 , , g n be a basis of a linear space V in which the quadratic form B(x, x) is written as B(x, x)= n i=1 ε i η 2 i ,(5.7.3.3) where η 1 , , η n are the coordinates of x in that basis, and ε 1 , , ε n are coefficients taking the values –1, 0,or1. Such a representation of a quadratic form is called its normal representation. Any real quadratic form B(x, x)inann-dimensional real linear space V admits a normal representation (5.7.3.3). Such a representation can be obtained by the following transformations: 1. One obtains its canonical representation (see Paragraph 5.7.3-4): B(x, x)= n i=1 λ i μ 2 i . 2. By the nondegenerate coordinate transformation η i = ⎧ ⎪ ⎨ ⎪ ⎩ 1 √ λ i μ i for λ i > 0, 1 √ –λ i μ i for λ i < 0, μ i for λ i = 0, the canonical representation turns into a normal representation. L AW OF INERTIA OF QUADRATIC FORMS. The number of terms with positive coefficients and the number of terms with negative coefficients in any normal representation of a real quadratic form does not depend on the method used to obtain such a representation. The index of inertia of a real quadratic form is the integer k equal to the number of nonzero coefficients in its canonical representation (this number coincides with the rank of the quadratic form). Its positive index of inertia is the integer p equal to the number of positive coefficients in the canonical representation of the form, and its negative index of inertia is the integer q equal to the number of its negative canonical coefficients. The integer s = p – q is called the signature of the quadratic form. A real quadratic form B(x, x)onann-dimensional real linear space V is a) positive definite (resp., negative definite) if p = n (resp., q = n); b) alternating if p ≠ 0, q ≠ 0; c) nonnegative (resp., nonpositive) if q = 0, p < n (resp., p = 0, q < n). 5.7. BILINEAR AND QUADRATIC FORMS 219 5.7.3-6. Criteria of positive and negative definiteness of a quadratic form. 1 ◦ . A real quadratic form B(x, x) is positive definite, negative definite, alternating, non- negative, nonpositive if the eigenvalues λ i of its matrix B ≡ [b ij ] are all positive, are all negative, some are positive and some negative, are all nonnegative, are all nonpositive, respectively. 2 ◦ . Sylvester criterion. A real quadratic form B(x, x) is positive definite if and only if the matrix of B(x, x)insomebasise 1 , , e n satisfies the conditions Δ 1 ≡ b 11 > 0, Δ 2 ≡ b 11 b 12 b 21 b 22 > 0, , Δ n ≡ det B > 0. If the signs of the minor determinants alternate, Δ 1 < 0, Δ 2 > 0, Δ 3 < 0, , then the quadratic form is negative definite. 3 ◦ . A real matrix B is nonnegative and symmetric if and only if there is a real matrix C such that B = C T C. 5.7.4. Bilinear and Quadratic Forms in Euclidean Space 5.7.4-1. Reduction of a quadratic form to a sum of squares. THEOREM 1. Let B(x, y) be a symmetric bilinear form on a n -dimensional Euclidean space V . Then there is an orthonormal basis i 1 , , i n in V and there are real numbers λ k such that for any x V the real quadratic form B(x, x) can be represented as the sum of squares of the coordinates ξ k of x in the basis i 1 , , i n : B(x, x)= n k=1 λ k ξ 2 k . T HEOREM 2. Let A(x, y) and B(x, y) be symmetric bilinear forms in a n -dimensional real linear space V , and suppose that the quadratic form A(x, x) is positive definite. Then there is a basis i 1 , , i n of V such that the quadratic forms A(x, x) and B(x, x) can be represented in the form A(x, x)= n k=1 λ k ξ 2 k , B(x, x)= n k=1 ξ 2 k , where ξ k are the coordinates of x in the basis i 1 , , i n . The set of real λ 1 , , λ n coincides with the spectrum of eigenvalues of the matrix B –1 A (the matrices A and B can be taken in any basis), and this set consists of the roots of the algebraic equation det(A – λB)=0. . on the reduction of a matrix to canonical Jordan form. 5.7. Bilinear and Quadratic Forms 5.7.1. Linear and Sesquilinear Forms 5.7.1-1. Linear forms in a unitary space. A linear form or linear functional. basic idea of the method consists of consecutive transformations of the quadratic form: on every step, one should single out the perfect square of some linear form. Consider a quadratic form B(x,. bilinear form that is polar to some positive definite quadratic form. The rank of a quadratic form on a finite-dimensional linear space V is, by definition, the rank of the matrix of that form in