na_lec_1

# na_lec_1 - 1 Singular Value Decomposition(SVD Unless stated...

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1. Singular Value Decomposition (SVD) Unless stated to the contrary, and are integers, and moreover . m n n m 1.1. The matrix , can be decomposed into n m × A T V U A Σ = where , m T T I UU U U = = { } n m i n i diag × = = Σ ,..., 2 , 1 , 0 σ , and . n T T I VV V V = = Moreover (a) The singular values n i i ,..., 2 , 1 , = are the square roots of the eigenvalues of and . T AA A A T (b) The columns of are eigenvectors of . U T AA (c) The columns of are eigenvectors of . V A A T Remark 1.1. The SVD do exist also for the case where n m < . But we will concentrate on the other case. 1.2. The compact form of the SVD is T V U A Σ = where n p is the number of nonzero eigenvalues of , and A A T { } , , ,..., 2 , 1 , 0 , , p T p n p p i m T p m p i diag I V V V I U U U = = = Σ = × × × . This result follows from the block-matrix multiplication rule. 1.3. With p as in 1.2 the SVD can be written in summation form = = p i T i i i 1 v u A where [] m u u u U L 2 1 = and [ ] n v v v V L 2 1 = . 1

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1.4.1. The pseudoinverse of a diagonal matrix D p i d d d d i n m p ,..., 2 , 1 , 0 ; 0 0 2 1 = = × O D O O is denoted by and given by + D () () () m n p d d d × + = 0 0 1 1 2 1 1 O O O D 1.4.2. The pseudoinverse of is denoted by and given by A + A (a) , or equivalently by m n T × + + Σ = U V A (b) , provided that
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na_lec_1 - 1 Singular Value Decomposition(SVD Unless stated...

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