7 Bilinear forms and inner products

Transcription

1 7 Bilinear forms and inner products Definition 7.1 A bilinear form θ on a vector space V over a field F is a function θ : V V F such that θ(λu+µv,w) = λθ(u,w)+µθ(v,w) θ(u,λv +µw) = λθ(u,v)+µθ(u,w) for all u,v,w V and all λ,µ F. We say that θ is linear in each argument. A bilinear form θ on V is called symmetric if for all u,v V. θ(u,v) = θ(v,u) Example 7.2 If A M n n (F) is any n n matrix with entries in the field F then there is a bilinear form θ A on the vector space F n over F defined by θ A (u,v) = uav T for all u,v F n. Moreover θ A is symmetric if and only if A is a symmetric matrix (that is, A T = A). For the first time in this course it is now important that our field of scalars F should be R, because we want to know what it means for scalars to be positive. Definition 7.3 A symmetric bilinear form θ on a vector space V over R is called positive definite if θ(u,u) 0 for all u V and θ(u,u) = 0 u = 0. A symmetric matrix A M n n (R) is called positive definite if the corresponding symmetric bilinear form θ A on R n defined in Example 7.2 above is positive definite. Example 7.4 The bilinear form θ In on R n associated as in Example 7.2 to the identity matrix I n M n n (R) is a positive definite symmetric bilinear form. θ In is the usual dot product on R n given by n θ In (u,v) = uv T = u.v = u i v i for all u = (u 1,...,u n ) and v = (v 1,...,v n ) in R n. Remark 7.5 It is traditional to use notation such as u.v or u,v (or even (u,v), though this is not recommended as it can be confused with the notation for ordered pairs) for positive definite symmetric bilinear forms (also called inner products) on real vector spaces. i=1 1

2 Definition 7.6 Let V be a real vector space. An inner product on V is a positive definite symmetric bilinear form on V; that is, it is a function, : V V R with the following properties. (1) For all v V we have v,v 0 and v,v = 0 v = 0 (positive definiteness). (2) For all v,w V and λ,µ R we have v,w = w,v (symmetry). (3)Forallu,v,w V wehave λu+µv,w = λ u,w +µ v,w (linearity in the first argument). Clearly linearity in the second argument follows from symmetry and linearity in the first argument. An inner product space is a real vector space V together with an inner product, on V. Example 7.7 As noted above, the usual dot product is an inner product on R n. So is θ A for any diagonal matrix A M n n (R) with strictly positive diagonal entries. Example 7.8 Let V be the vector space R n [X] of polynomials of degree at most n in X with real coefficients. Then there is an inner product, on V given by f,g = 1 0 f(t)g(t)dt. An inner product allows us to define the length of a vector. Definition 7.9 Let, be an inner product on a real vector space V. If v V then the length of v is v = v,v. Theorem 7.10 Cauchy-Schwartz inequality Let, be an inner product on a real vector space V. If u,v V then u,v u v with equality holding if and only if u and v are linearly dependent (that is, one is a scalar multiple of the other). Proof: If u,v V and λ R it follows from the properties of an inner product that 0 u λv,u λv = u,u 2λ u,v +λ 2 v,v = u 2 2λ u,v +λ 2 v 2 with equality if and only if u = λv. If u = 0 or v = 0 then the result is trivial, so we can assume that u 0 and v 0. Then we can take λ = ± u / v to get 2 u 2 ±2 u v 1 u,v and hence u,v u v. Moreover if equality holds here then u = λv so u and v are linearly dependent, and conversely if one of u and v is a scalar multiple of the other then it is easy to check that equality holds. The Cauchy-Schwartz inequality allows us to define the angle between two non-zero vectors in an inner product space. 2

3 Definition 7.11 Let, be an inner product on a real vector space V. The angle between non-zero vectors u and v in V is the unique element φ of [0,π] such that u,v = u v cosφ. Vectors u,v V are said to be orthogonal if u,v = 0. Remark 7.12 We can also define inner products (often called Hermitian inner products) for complex vector spaces, but we need to modify the axioms for real inner products, since they become inconsistent over the complex numbers. Definition 7.13 Let V be a complex vector space. A Hermitian inner product on V is a function, : V V C with the following properties. (1) For all v V we have v,v R and v,v 0, and v,v = 0 v = 0 (positive definiteness). (2) For all v,w V and λ,µ R we have v,w = w,v (complex-conjugate symmetry). (3)Forallu,v,w V wehave λu+µv,w = λ u,w +µ v,w (linearity in the first argument). From complex-conjugate symmetry and linearity in the first argument we get complexconjugate linearity in the second argument: u,λv+µw = λ u,v + µ u,w for all u,v,w V. A Hermitian inner product space is a vector space V over C together with a Hermitian inner product, on V. Remark 7.14 The length of a vector in a Hermitian inner product space can be defined in exactly the same way as in a real vector space, and the definition of orthogonality of two vectors is also the same; however there is no analogous definition of the angle between two non-zero vectors. The Cauchy-Schwartz inequality for a Hermitian inner product space says that for all vectors u and v u,v u v with equality holding if and only if u and v are linearly dependent (the proof is the same as in the real case except that λ has to be chosen more carefully). Definition 7.15 Let, be an inner product on a real vector space V. A subset B of V is said to be orthogonal if u,v = 0 for all u,v B such that u v. It is said to be orthonormal if in addition u,u = 1 for all u B. An orthonormal basis of V is an orthonormal subset of V which is also a basis of V. Lemma 7.16 Let, be an inner product on a real vector space V and let B be an orthogonal subset of V consisting of non-zero vectors. Then B is linearly independent. Proof: Suppose that v 1,...,v n are distinct elements of B and λ 1,...,λ n R and that λ 1 v 1 + +λ n v n = 0. Then if 1 j n we have 0 = λ 1 v 1 + +λ n v n,v j = λ 1 v 1,v j + +λ n v n,v j = λ j v j,v j. By the hypothesis on B we have v j 0 and so v j,v j 0, and therefore λ j = 0. Thus B is linearly independent. 3

4 Lemma 7.17 Let, be an inner product on a finite-dimensional real vector space V. Let B = {v 1,...v n } be a basis for V, and for each v V let [v] B be the coordinate vector of v with respect to the basis B. Then the basis B is orthonormal if and only if Proof to be completed. u,v = ([u] B ) T [v] B for all u,v V. Recall that a square real matrix A is orthogonal if and only if A 1 = A T. Definition 7.18 Let, V be an inner product on a real vector space V and let, W be an inner product on a real vector space W. A linear transformation T : V W is called an orthogonal linear transformation of the inner product spaces if T(u),T(v) W = u,v V for all u,v V. Lemma 7.19 A linear transformation T : V W of real inner product spaces is orthogonal if and only its matrix A with respect to any orthonormal bases B V and B W of V and W is orthogonal. Proof: If v V let [v] BV be the coordinate vector of v with respect to the basis B V, and if w W let [w] BW be the coordinate vector of w with respect to the basis B W. Then if v V we have [T(v)] BW = A[v] BV and so by Lemma 7.17 if u,v V then T(u),T(v) W = ([T(u)] BW ) T [T(v)] BW = (A[u] BV ) T A[v] BV = ([u] BV ) T A T A[v] BV. If A is orthogonal then this is equal to [u] T B V [v] BV = u,v and so T is orthogonal. Conversely if T is orthogonal then ([u] BV ) T A T A[v] BV = [u] T B V [v] BV for all u,v V, and so taking u and v to be i-th and j-th elements of the orthonormal basis B V we find that the (i,j)-th entry of A T A is the (i,j)-th entry of the identity matrix for all choices of i and j, and thus A is orthogonal. Remark 7.20 (looking ahead to Linear Algebra II) The Gram Schmidt procedure allows us to construct from any basis {v 1,...,v n } of a finitedimensional real inner product space V an orthonormal basis {e 1,...,e n } such that Sp({e 1,...,e k }) = Sp({v 1,...,v k }) for k = 1,...,n, as follows. Let e 1 = v 1 / v 1, which is well defined since {v 1,...,v n } is linearly independent and so v 1 0. Then e 1 = 1. 4

5 Let e 2 = w 2 / w 2 where w 2 = v 2 v 2,e 1 e 1. This is well defined since {v 1,...,v n } is linearly independent and e 1 is a scalar multiple of v 1 and so w 2 0. Then e 2 = 1 and e 1,e 2 = 0 and Sp({e 1,e 2 }) = Sp({v 1,v 2 }). Assume inductively that we have defined an orthonormal set {e 1,...,e k } such that Sp({e 1,...,e j }) = Sp({v 1,...,v j }) for j = 1,...,k. Let e k+1 = w k+1 / w k+1 where w k+1 = v k+1 v k+1,e 1 e 1 v k+1,e k e k. Thisiswelldefinedsince{v 1,...,v n }islinearlyindependentandsp({e 1,...,e k }) = Sp({v 1,...,v k }) and so w k+1 0. Then e k+1 = 1 and e j,e k+1 = 0 if 1 j k and Sp({e 1,...,e k+1 }) = Sp({v 1,...,v k+1 }) by the Steinitz Exchange Lemma. We can repeat this procedure until we obtain an orthonormal basis {e 1,...,e n } for V such that Sp({e 1,...,e k }) = Sp({v 1,...,v k }) for k = 1,...,n. Notealsothatif{v 1,...,v k }isalreadyorthonormalforsomek thenwewillgete j = v j forj = 1,...,k. So if S = {e 1,...e k } is any orthonormal subset of V, then S is linearly independent by Lemma 7.16, so we can extend S to a basis {e 1,...,e k,v k+1,...,v n } for V and then apply the Gram Schmidt procedure to obtain an orthonormal basis {e 1,...,e k,e k+1,...,e n } for V containing S. 5