solhw4

solhw4 - Solutions for Homework 4 Numerical Linear Algebra...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solutions for Homework 4 Numerical Linear Algebra 1 Fall 2010 Problem 4.1 Given that we know the SVD exists for any complex matrix A C m n , assume that A R m n has rank k with k n , i.e., A is real and it may be rank deficient, and show that the SVD of A is all real and has the form A = U S V T = U k k V T k where S R n n is diagonal with nonnegative entries, U = ( U k U m- k ) , U T U = I m V = ( V k V n- k ) , V T V = I n U k R m k , and V k R n k Solution: The result follows simply from the relationship of the SVD to the symmetric eigenvalue decomposition. A T A and AA T are both real symmetric positive semidefinite and therefore have real orthogonal eigenvectors that define orthogonal U and V as needed. The rank is reflected in the nonzero eigenvalues which are the squares of the nonzero singular values and therefore define k and S . U k and V k follow immediately from the appropriate partitionings of U and V . Problem 4.2 Let T R n n be a symmetric tridiagonal matrix, i.e., e T i Te j = e T j Te i and e T i Te j = 0 if j < i- 1 or j > i + 1. Consider T = QR where R R n n is an upper triangular matrix and Q R n n is an orthogonal matrix. Recall, the nonzero structure of R was derived in class and shown to be e T i Re j = 0 if j < i (upper triangular assumption) or if j > i + 2, i.e, nonzeros are restricted to the main diagonal and the first two superdiagonals. (4.2.a) Show that Q has nonzero structure such that e T i Qe j = 0 if j < i- 1, i.e., Q is upper Hessenberg. (4.2.b) Show that T + = RQ is a symmetric triagonal matrix. (4.2.c) Prove the Lemma in the class notes that states that choosing the shift = , where is an eigenvalue of T , results in a reduced T + with known eigenvector and eigenvalue....
View Full Document

Page1 / 5

solhw4 - Solutions for Homework 4 Numerical Linear Algebra...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online