HW6_2011 - 1 ,..., r of M * M . Denote by u j the columns...

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Math 602 Homework 6 1. Show that the pseudoinverse of a vector x C n is x + = ± 0 , 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 = ² 1 1 - 2 - 2 ³ L = ² 10 2 10 2 5 11 5 11 ³ 3. The null space of a matrix is also called its kernel Ker( M ) = N ( M ) = { x | Mx = 0 } . Recall that Ran( M ) = { y | y = Mx for some x } . Here are some very useful facts; make note of them for future use. Prove the following (the proofs are very simple!) : (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) Ran( MM * ) = Ran( M ) (ix) Ran( M * M ) = Ran( M * ) Important remark for infinite dimensions: we shall see that while (iv) , (v) , (viii) , (ix) need a slight modification, everything else is true in infinite dimensions as well. Continued on next page. .. 1
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4. Here are some facts about SVD which may be useful. Let M = U Σ V * be the SVD of the matrix M . Denote by r the number of the nonzero eigenvalues
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Unformatted text preview: 1 ,..., r of M * M . Denote by u j the columns of U and by v j the columns of V , where M * Mv j = j v j for j = 1 ,...,r . Prove the following (you may wish to revisit the proof of the SVD, and to use properties stated in problem 3.): (i) The span of v r +1 ,v r +2 ,... is Ker( M ) (=the nullspace of M ). (ii) The span of u 1 ,...,u r is Ran( M ) (=the column space of M ). (iii) The span of u r +1 ,u r +2 ,... is Ker( M * ) (=the left nullspace of M ) (iv) The span of v 1 ,...,v r is Ran( M * ) (=the row space of M ). 5. Prove the following properties of the pseudoinverse M + of the matrix M (you may wish to use properties listed in problem 4.): (i) MM + is the orthogonal projector on Ran( M ). (ii) M + M is the orthogonal projector on Ker( M ) . 2...
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HW6_2011 - 1 ,..., r of M * M . Denote by u j the columns...

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