This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Homework 5 Numerical Linear Algebra 1 Fall 2011 Due date: beginning of class Wednesday, 11/9/11 Problem 5.1 Let A R n k have rank k . The pseudoinverse for rectangular full columnrank matrices behaves much as the inverse for nonsingular matrices. To see this show the following identities are true (Stewart 73): 5.1.a . AA A = A 5.1.b . A AA = A 5.1.c . A A = ( A A ) T 5.1.d . AA = ( AA ) T 5.1.e . If A R n k has orthonormal columns then A = A T . Why is this important for consistency with simpler forms of least squares problems that we have discussed? Problem 5.2 Any subspace S of R n of dimension k n must have at least one orthogonal matrix Q R n k with orthonormal columns such that R ( Q ) = S , The matrix P = QQ T is a projector onto S , i.e., Px is the unique component of x contained in S . 5.2.a . P is clearly symmetric, show that it is idempotent, i.e., P 2 = P ....
View Full
Document
 Fall '06
 gallivan

Click to edit the document details