- #1

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In a given matrix A, the singular value decomposition (SVD), yields A=USV`. Now lets make dimension reduction of the matrix by keeping only one column vector from U, one singular value from S and one row vector from V`. Then do another SVD of the resulted rank reduced matrix Ar.

Now, if Ar is the result of multiplication of Ur , Sr and V`r, then why the result, shown in the right picture in the attached doc, still has non-vanishing columns of Ur and non-vanishing rows of V`r? in other words, where do Ur1, Ur2, V`r1 and V`r2 come from as long as other values of S, namely S2 and S3 are zero?

Now, if Ar is the result of multiplication of Ur , Sr and V`r, then why the result, shown in the right picture in the attached doc, still has non-vanishing columns of Ur and non-vanishing rows of V`r? in other words, where do Ur1, Ur2, V`r1 and V`r2 come from as long as other values of S, namely S2 and S3 are zero?