3 Orthogonal Vectors and Matrices


 Claude Gyles Clark
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1 3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first two of these factorizations involve orthogonal matrices These matrices play a fundamental role in many numerical methods This lecture first considers orthogonal vectors and then defines orthogonal matrices 3 Orthogonal Vectors A pair of vector u,v R m is said to be orthogonal if (u,v) = 0 In view of formula () in Lecture, orthogonal vectors meet at a right angle The zerovector 0 is orthogonal to all vector, but we are more interested in nonvanishing orthogonal vectors A set of vectors S n = {v j } n in Rm is said to be orthonormal if each pair of distinct vectors in S n is orthogonal and all vectors in S n are of unit length, ie, if (v j,v k ) = { 0, j k,, j = k () Here we have used that (v k,v k ) = v k It is not difficult to show that orthonormal vectors are linearly independent; see Exercise 3 below It follows that the m vectors of an orthonormal set S m in R m form a basis for R m Example 3 The set S 3 = {e j } 3 in R5 is orthonormal, where the e j are axis vectors; cf (5) of Lecture Example 3 The set S = {v,v } in R, with v = [,] T, v = [,] T, is orthonormal Moreover, the set S forms a basis for R An arbitrary vector v R m can be decomposed into orthogonal components Consider the set S n = {v j } n of orthonormal vectors in Rm, and regard the expression r = v (v j,v)v j () The vector (v j,v)v j is referred to as the orthogonal component of v in the direction v j Moreover, the vector r is orthogonal to the vectors v j This can be seen by computing the inner products (v k,r) for all k We obtain (v k,r) = (v k,v (v j,v)v j ) = (v k,v) (v j,v)(v k,v j )
2 Using (), the sum in the righthand side simplifies to which shows that (v j,v)(v k,v j ) = (v k,v), (v k,r) = 0 Thus, v can be expressed as a sum of the orthogonal vectors r,v,,v n, Example 33 v = r + (v j,v)v j Let v = [,,3,4,5] T and let the set S be the same as in Example 3 Then r = [0,0,0,4,5] T Clearly, r is orthogonal to the axis vectors e,e,e 3 Exercise 3 Let S n = {v j } n be a set of orthonormal vectors in Rm Show that the vectors v,v,,v n are linearly independent Hint: Assume this is not the case For instance, assume that v is a linear combination of the vectors v,v 3,,v n, and apply () 3 Orthogonal Matrices A square matrix Q = [q,q,,q m ] R m m is said to be orthogonal if its columns {q j } m form an orthonormal basis of R m Since the columns q,q,,q m are linearly independent, cf Exercise 3, the matrix Q is nonsingular Thus, Q has an inverse, which we denote by Q It follows from the orthonormality of the columns of Q that (q,q ) (q,q ) (q,q m ) Q T (q,q ) (q,q ) (q,q m ) Q = = I, (q m,q ) (q m,q ) (q m,q m ) where I denotes the identity matrix Multiplying the above expression by the inverse Q from the righthand side shows that Q T = Q Thus, the transpose of an orthogonal matrix is the inverse Example 34 The identity matrix I is orthogonal
3 Example 35 The matrix is orthogonal Its inverse is its transpose, [ Q = Q = Q T = / / / / ] [ / / / / Geometrically, multiplying a vector by an orthogonal matrix reflects the vector in some plane and/or rotates it Therefore, multiplying a vector by an orthogonal matrices does not change its length Therefore, the norm of a vector u is invariant under multiplication by an orthogonal matrix Q, ie, ] Qu = u (3) This can be seen by using the properties (8) and (6) of Lecture We have Qu = (Qu) T (Qu) = u T Q T (Qu) = u T (Q T Q)u = u T u = u We take squareroots of the righthand side and lefthand side, and since is nonnegative, equation (3) follows 33 Householder Matrices Matrices of the form H = I ρuu T R m m, u 0, ρ = u T u, (4) are known as Householder matrices They are used in numerical methods for leastsquares approximation and eigenvalue computations We will discuss the former application in the next lecture Householder matrices are symmetric, ie, H = H T, and orthogonal The latter property follows from H T H = H = (I ρuu T )(I ρuu T ) = I ρuu T ρuu T + (ρuu T )(ρuu T ) = I ρuu T ρuu T + ρu(ρu T u)u T = I ρuu T ρuu T + ρuu T = I, where we have used that ρu T u = ; cf (4) Our interest in Householder matrices stems from that they are orthogonal and the vector u in their definition can be chosen so that an arbitrary (but fixed) vector w R m \{0} is mapped by H onto a multiple of the axis vector e We will now show how this can be done Let w 0 be given We would like for some scalar σ It follows from (3) that Hw = σe (5) w = Hw = σe = σ e = σ Therefore, σ = ± w (6) 3
4 Moreover, using the definition (4) of H, we obtain from which it follows that σe = Hw = (I ρuu T )w = w τu, τu = w σe τ = ρu T w The matrix H is independent of the scaling factor τ in the sense that the entries of the matrix H do not change if we replace τu by u We therefore may choose u = w σe (7) This choice of u and either one of the choices (6) of σ give a Householder matrix that satisfies (5) Nevertheless, in finite precision arithmetic, the choice of sign in (6) may be important To see this, let w = [w,w,,w m ] T and write the vector (7) in the form u = w σ w w 3 w m If the components w j for j are of small magnitude compared to w, then w w and, therefore, the first component of u satisfies u = w σ = w ± w w ± w (8) We would like to avoid the situation that w is large and u is small, because then u is determined with low relative accuracy; see Exercise 33 below We therefore let which yields Thus, u is computed by adding numbers of the same sign Example 36 σ = sign(w ) w, (9) u = w + sign(w ) w (0) Let w = [,,,] T We are interested in determining the Householder matrix H that maps w onto a multiple of e The parameter σ in (5) is chosen, such that σ = w =, ie, σ is or The vector u in (4) is given by u = w ± σe = The choice σ = yields u = [3 ] T We choose the σ positive, because the first entry of w is positive Finally, we obtain ρ = / u = /6 and / / / / H = I ρuu T = / 5/6 /6 /6 / /6 5/6 /6 / /6 /6 5/6 ± σ 4
5 It is easy to verify that H is orthogonal and maps w to e In most applications it is not necessary to explicitly form the Householder matrix H Matrixvector products with a vector v are computed by using the definition (4) of H, ie, Hv = (I ρuu T )v = v (ρu T v)u The lefthand side is computed by first evaluating the scalar τ = ρu T v and then computing the vector scaling and addition v τu This way of evaluating Hw requires fewer arithmetic floatingpoint operations than straightforward computation of the matrixvector product; see Exercise 36 Moreover, the entries of H do not have to be stored, only the vector u and scalar ρ The savings in arithmetic operations and storage is important for large problems Exercise 3 Let w = [,,3] T Determine the Householder matrix that maps w to a multiple of e Only the vector u in (4) has to be computed Exercise 33 This exercise illustrates the importance of the choice of the sign of σ in (6) Let w = [, ] T and let u be the vector in the definition of Householder matrices (4), chosen such that Hw = σe MATLAB yields >> w=[; 5e9] w = >> sigma=norm(w) sigma = >> u=w()+sigma u = >> u=w()sigma u = 0 5
6 where the u denote the computed approximations of the first component, u, of the vector u How large are the absolute and relative errors in the computed approximations u of the component u of the vector u? Exercise 34 Show that the product U U of two orthogonal matrices is an orthogonal matrix Is the product of k > orthogonal matrices an orthogonal matrix? Exercise 35 Let Q be an orthogonal matrix, ie, Q T Q = I Show that QQ T = I Exercise 36 What is the count of arithmetic floating point operations for evaluating a matrix vector product with an n n Householder matrix H when the representation (4) of H is used? Only u and ρ are stored, not the entries of H What is the count of arithmetic floating point operations for evaluating a matrixvector product with H when the entries of H (but not u and ρ) are available Correct order of magnitude of the arithmetic work when n is large suffices Exercise 37 Let the nonvanishing mvectors w and w be given Determine an orthogonal matrix H, such that w = Hw Hint: Use Householder matrices 6
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