Linear Algebra expression Let A be an mxn matrix (assume m > n). The full singul

Question

Linear Algebra expression

Let A be an mxn matrix (assume m > n). The full singular value factorization A = U V^T contains more information than necessary to reconstruct A. (a) What are the smallest matrices U, and V_T such that U V = A? (b) Let U = [ U U ]. That is, think about U from the full singular value factorization as a block matrix consisting of the matrix U found in part (a) and the remaining (unneeded) columns U. Find expressions for IFU and UUT. (c) Use the reduced singular value factorization obtained in part (a) to find an expression for the matrix H = A(A^TA)-lA^T. How many matrices must be inverted (diagonal and orthogonal matrices don't count)?

Explanation / Answer

a) U = [u1 u2 u3.... um], m x n matrix

   = n x n diagonal matrix

   V = [v1 v2 v3... vn] is n x n

This is because, since m > n, we need only n EigenValues and the remaining m - n zeroes are redundant.

b) UTU and UUT both = I

c) ATA= (UTV).UVT = (2)-1

   A* above expresion gives = UV'*2)-1 = U-1VT

Above expression * AT= U-1VT * UTV = UUT*-1VTV = I

Only the diagonal matrix needs to be inverted.

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