Lecture #14 Notes - Lecture 14 Gram-Schmidt Process QR...

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Lecture 14: Gram-Schmidt Process, QR Factorization and OrthogonalTransformationsGoal:Gram-Schmidt process and QR factorization.Orthogonal transformations, orthogonal matrices, matrix transpose and the matrix of an orthog-onal projection.1. Gram-Schmidt ProcessConsider a basis~v1, . . . ,~vmof a subspaceVofRn. LetV1= span(~v1),V2= span(~v1,~v2),...Vm= span(~v1, . . . ,~vm) =V.To find an orthonormal basis ofV1, we normalize the basis vector~v1and get~u1=~v1k~v1kwhich forms an orthonormal basis ofV1.To find an orthonormal basis ofV2, we have~u1V2and look for a unit vector~u2which is orthogonalto~u1such that span(~u1, ~u2) =V2. Now we decompose~v2:~v2=~vk2+~v2= projV1(~v2) +~v2where~v2V1.Since~v2=~v2-~vk2V2and~v2~u1implying that~v2and~u1are linearly independent,~v2and~u1form a basis ofV2. Let~u2=~v2~v2=~v2-(~v2·~u1)~u1k~v2-(~v2·~u1)~u1k.Then~u1, ~u2form an orthonormal basis ofV2.Similarly, at thej-th step, assume that we get{~u1, ~u2, . . . , ~uj-1}as an orthonormal basis ofVj-1. Let~uj=~vjk~vjk=~vj-(~vj·~u1)~u1-(~vj·~u2)~u2- · · · -(~vj·~uj-1)~uj-1k~vj-(~vj·~u1)~u1-(~vj·~u2)~u2- · · · -(~vj·~uj-1)~uj-1k.Then we can prove that~u1, . . . , ~ujform an orthonormal basis ofVj.2. QR FactorizationLetrij=~ui·~vjfori6=jandrjj=k~vjk. We have~vj=r1j~u1+r2j~u2+· · ·+rj-1,j~uj-1+rjj~uj=||· · ·|~u1~u2· · ·~um||· · ·|r1j...rjj0...0.1

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