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5.1 Orthogonality Definitions A set of vectors is called an orthogonal set if all pairs of distinct vectors in the set are orthogonal. An orthonormal set is an orthogonal set of unit vectors. An orthogonal (orthonormal) basis for a subspace W of n R is a basis for W that is an orthogonal (orthonormal) set. An orthogonal matrix is a square matrix whose columns form an orthonormal set. Examples 1) Is the following set of vectors orthogonal? orthonormal? a) 1 2 3 2 1 , 4 , 1 1 2 b) { e1 , e 2 ,..., e n } 2) Find an orthogonal basis and an orthonormal basis n for the subspace W of R x W y : x y 2 z 0 z Theorems All vectors in an orthogonal set are linearly independent. Let {v1, v2,…, vk } be an orthogonal basis for a subspace n W of R and w be any vector in W. Then the unique scalars c1 ,c2 , …, ck such that w = c1v1 + c2v2 + …+ ckvk are given by w vi ci for i 1,..., k vi vi Proof: To find ci we take the dot product with vi w vi = (c1v1 + c2v2 + …+ ckvk ) vi Examples 3) The orthogonal basis for the subspace W in previous example is 1 1 1 , 1 0 1 Pick a vector in W and express it in terms of the vectors in the basis. 4) Is the following matrix orthogonal? 3 A 1 2 2 4 1 1 1 2 0 B 1 0 0 0 1 1 0 0 cos C sin sin cos If it is orthogonal, find its inverse and its transpose. Theorems on Orthogonal Matrix The following statements are equivalent for a matrix A : 1) A is orthogonal -1 T 2) A = A n 3) ||Av|| = ||v|| for every v in R n 4) Av1∙ Av2 = v1∙ v2 for every v1 ,v2 in R Let A be an orthogonal matrix. Then 1) its rows form an orthonormal set. -1 2) A is also orthogonal. 3) |det(A)| = 1 4) |λ| = 1 where λ is an eigenvalue of A 5) If A and B are orthogonal matrices, then so is AB 5.2 Orthogonal Complements and Orthogonal Projections Orthogonal Complements Recall: A normal vector n to a plane is orthogonal to every vector in that plane. If the plane passes through the origin, then it is a subspace W of R3 . Also, span(n) is also a subspace of R3 Note that every vector in span(n) is orthogonal to every vector in subspace W . Then span(n) is called orthogonal complement of W. Definition: A vector v is said to be orthogonal to a subspace W of n R if it is orthogonal to all vectors in W. The set of all vectors that are orthogonal to W is called the orthogonal complement of W, denoted W ┴ . That is W perp W {v R n : v w 0 w W} http://www.math.tamu.edu/~yvorobet/MATH304-2011C/Lect3-02web.pdf Example 3 1) Find the orthogonal complements for W of R . 1 a) W span 2 3 b) W plane with direction c) (subspace x W y : x y 2 z 0 z spanned by) vectors 1 0 1 and 1 0 1 Theorems n Let W be a subspace of R . n ┴ 1) W is a subspace of R . 2) (W ┴)┴ = W 3) W ∩ W ┴ = {0} 4) If W = span(w1,w2,…,wk), then v is in W ┴ iff v∙wi = 0 for all i =1,…,k. Let A be an m x n matrix. Then (row(A))┴ = null(A) and (col(A))┴ = null(AT) Proof? Example 2) Use previous theorem to find the orthogonal complements 3 for W of R . 1 0 a) W plane w ith direction (subspace spann ed by) vectors 1 and 1 0 1 b) 3 2 W subspace spanned by vectors 0 , 1 4 1 2 2 an d 0 1 3 2 6 2 5 Orthogonal Projections u v w 1 proj v u v 2 u w2 w1 u v v v v v w 2 = perp v u u - w 1 v Let u and v be nonzero vectors. w1 is called the vector component of u along v (or projection of u onto v), and is denoted by projvu w2 is called the vector component of u orthogonal to v Orthogonal Projections n Let W be a subspace of R with an orthogonal basis {u1, u2,…, uk }, the orthogonal projection of v onto W is defined as: projW v = proju v + proju v + … + proju v 1 2 k The component of v orthogonal to W is the vector perpW v = v – projw v n n Let W be a subspace of R and v be any vector in R . Then there are unique vectors w1 in W and w2 in W ┴ such that v = w1 + w2 . Examples 3) Find the orthogonal projection of v = [ 1, -1, 2 ] onto W and the component of v orthogonal to W. 1 a) W span 2 3 1 -1 b) W subspace spanned by 1 and 1 0 1 x c) W y : x y 2 z 0 z 5.3 The Gram-Schmidt Process And the QR Factorization The Gram-Schmidt Process Goal: To construct an orthogonal (orthonormal) basis for n any subspace of R . We start with any basis {x1, x2,…, xk }, and “orthogonalize” each vector vi in the basis one at a time by finding the component of vi orthogonal to W = span(x1, x2,…, xi-1 ). Let {x1, x2,…, xk } be a basis for a subspace W. Then choose the following vectors: v1 = x1, v2 = x2 – projv x2 1 v3 = x3 – projv x3 – projv x3 1 2 … and so on Then {v1, v2,…, vk } is orthogonal basis for W . We can normalize each vector in the basis to form an orthonormal basis. Examples 1) Use the following basis to find an orthonormal basis for R 3 1 , , 1 2 3 2) Find an orthogonal basis for R that contains the vector 1 1 1 2 , 1 , 0 1 1 1 2 The QR Factorization If A is an m x n matrix with linearly independent columns, then A can be factored as A = QR where R is an invertible upper triangular matrix and Q is an m x n orthogonal n matrix. In fact columns of Q form orthonormal basis for R which can be constructed from columns of A by using Gram-Schmidt process. Note: Since Q is orthogonal, Q-1 = QT and we have R = QT A Examples 3) Find a QR factorization for the following matrices. 3 A 1 1 2 1 A 2 1 -1 1 -1 - 1 0 1 5.4 Orthogonal Diagonalization of Symmetric Matrices Example 1) Diagonalize the matrix. 3 A 2 2 6 Recall: A square matrix A is symmetric if AT = A. A square matrix A is diagonalizable if there exists a matrix P and a diagonal matrix D such that P-1AP = D. Orthogonal Diagonalization Definition: A square matrix A is orthogonally diagonalizable if there exists an orthogonal matrix Q and a diagonal matrix D such that Q-1AQ = D. Note that Q-1 = QT Theorems 1. If A is orthogonally diagonalizable, then A is symmetric. 2. If A is a real symmetric matrix, then the eigenvalues of A are real. 3. If A is a symmetric matrix, then any two eigenvectors corresponding to distinct eigenvalues of A are orthogonal. A square matrix A is orthogonally diagonalizable if and only if it is symmetric. Example 2) Orthogonally diagonalize the matrix 0 A 1 1 1 0 1 1 1 0 and write A in terms of matrices Q and D. Theorem If A is orthogonally diagonalizable, and QTAQ = D then A can written as A 1 q1 q1 2 q 2 q 2 ... n q n q n T T T where qi is the orthonormal column of Q, and λi is the corresponding eigenvalue. This fact will help us construct the matrix A given eigenvalues and orthogonal eigenvectors. Example 3) Find a 2 x 2 matrix that has eigenvalues 2 and 7, with corresponding eigenvectors v1 1 2 v 2 1 2 5.5 Applications Quadratic Forms A quadratic form in x and y : a T 2 2 ax by cxy x 1 2 c c x b 1 2 A quadratic form in x,y and z: a 2 2 2 ax by cz dxy exz fyz x T 12 d 12 e where x is the variable (column) matrix. 1 2 d b 1 2 f e 1 f x 2 c 1 2 Quadratic Forms A quadratic form in n variables is a function n f : R R of the form: f (x) x Ax T where A is a symmetric n x n matrix and x is in R A is called the matrix associated with f. z f ( x , y ) x y 8 xy 2 2 z f ( x, y ) 2 x 5 y 2 2 n The Principal Axes Theorem Every quadratic form can be diagonalized. In fact, if A is a symmetric n x n matrix and if Q is an orthogonal matrix so that QTAQ = D then the change of variable x = Qy transforms the quadratic form into x A x y D y 1 y1 2 y 2 ... n y n T T 2 2 2 Example: Find a change of variable that transforms the Quadratic into one with no cross-product terms. z f ( x , y ) x y 8 xy 2 2 z f ( x, y ) 2 x 5 y 2 2