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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 column-rank 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 ....
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- Fall '06
- Linear Algebra, Diagonal matrix, Gram Schmidt, Classical Gram Schmidt