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homework04 - Homework 4 Stat 541 Due Friday 12 noon Linear...

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Homework 4, Stat 541: Due Friday, Oct 16, 2009, 12 noon Linear Algebra and Linear Models Student Name: (replace this with your name) October 6, 2009 Instructions: Edit this LaTex file with your solutions and generate a PDF file from it. E-mail the PDF to the usual class gmail address. For some questions one needs to understand basis changes and associated coordinate transfor- mations. To brush up (or to finally really understand what that is), you may want to check the solutions of Homework 2, Problem 11. See also Strang’s text on linear algebra, the Appendix ”Linear Transformations, Matrices, and Change of Basis.” The usual honor code applies (see the class webpage). In particular, it is strictly prohibited to consult previous years’ solutions. 1. Interpretation of eigendecompositions: You know that for any real symmetric p × p matrix S there exists an orthonormal basis ( u j ) j =1 ...p of eigenvectors and associated eigenvalues ( λ j ) j =1 ...p which can be assumed in descending order w.l.o.g.: λ 1 λ 2 ... λ p . State what it means for u j to be an eigenvector with eigenvalue λ j , both formally (state the definition) and geometrically (how does an eigenvector move under S ?). Comment on what it means if λ j = 1 or λ j = - 1 or λ j = 0. Answer: 2. Under the assumption that the vectors u j form an orthonormal system of eigenvectors of S , form the matrix U = ( u 1 , ..., u p ) and form the diagonal matrix Λ that has the eigenvalues in the diagonal (zero off-diagona). Express what orthonormality of the vectors u j means 1
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