Suppose the m x n matrix A is decomposed in the form 2 0 , A _ U ( 0 0 ) V Where U and V are unitary matrices, and 2 is an invertible 'r X 1* matrix...

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3. Suppose the m x n matrix A is decomposed in the form 2 0 ,A _ U ( 0 0 ) VWhere U and V are unitary matrices, and 2 is an invertible 'r X 1* matrix ( the SVD could be used to produce such a decomposition). The ”Moore-Penrose inverse” denoted byA+, can be defined as the 'n, x m matrix 2‘1 0+_ IA —V( 0 0)U (you can invoke it in Matlab using pinv(A)) a) Show that A+A and AA+ are symmetric, and that AA+A = A and A+AA+ = A". b) Show that, of all :I: that minimize ||y — Aa:| |2 , the one with the smallest length ||:L'| |2is given by 5?: = A+y.