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HW6_2012

# HW6_2012 - Math 602 Homework 6 1 Show that the...

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Unformatted text preview: Math 602 Homework 6 1. Show that the pseudoinverse of a vector x ∈ C n is x + = , if x = 0 1 k x k 2 x * , if x 6 = 0 2. Find the SVD and the pseudoinverse of the following matrices: M = 1 1- 2- 2 L = 10 2 10 2 5 11 5 11 3. Let u 1 ,...,u r be an orthonormal set in C n (with r ≤ n ) and consider the matrix R whose columns are these vectors: R = [ u 1 ,...,u r ]. Show that R * R and RR * are projectors (in general none of them equals I ) and find the SVD and the pseudoinverse of R . (You may wish to use the statements of Problems 4 and 5 below.) 4. Here are some very useful and general facts. Make note of them for future use. (Their proofs are very simple!) Recall that Ker( M ) = { x | Mx = 0 } and Ran( M ) = { y | y = Mx for some x } . Prove the following: (i) If h y,x i = 0 for all x then y = 0. (ii) Ker( M * ) = Ran( M ) ⊥ (iii) Ker( M ) = Ran( M * ) ⊥ (iv) Ker( M * ) ⊥ = Ran( M ) (v) Ker( M ) ⊥ = Ran( M * ) (vi) Ker( M * M ) = Ker( M ) (vii) Ker( MM * ) = Ker( M * ) (viii)...
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HW6_2012 - Math 602 Homework 6 1 Show that the...

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