svd:  A=U S V'    with S of the same shape as A has a lot of advantages:

1) It is the ultimate tool to solve linear least squares problems : it is numerically robust and gives you good information how much to regularize the problem in case of singularity or illconditioning  (by replacing zero or small singular values by some reasonable positive value for example in the range of uncertainty of elements of A.)

2) It provides the solution of minimal norm in case of nonuniqueness in the case of singular least squares problems ||Ax-b||_2 = min :

  x=VS'U'b

3) in the case of a homogeneous linear system with rank deficiency 1 it provides directly the (up to normalization unique) solution as the column of V which corresponds to the zero singular value  (e.g. fitting a plane in 3d from data points)

4) It gives you the exact l2-condition number opposed to condition estimators available with other decompositions as max s_{ii}/( min s_{ii} : s_{ii}>0 )

5) It gives you the Moore-Penrose pseudoinverse of A  (as VS'U' )

1. It provides the solution of minimal norm in case of nonuniqueness in the case of singular least squares problems ||Ax-b||_2 = min :

x=VS'U'b

2. in the case of a homogeneous linear system with rank deficiency 1 it provides directly the (up to normalization unique) solution as the column of V which corresponds to the zero singular value (e.g. fitting a plane in 3d from data points)

3. It gives you the exact l2-condition number opposed to condition estimators available with other decompositions as max s_{ii}/( min s_{ii} : s_{ii}>0 )

4. It gives you the Moore-Penrose pseudoinverse of A (as VS'U' )

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