Let $A$ and $B$ be any real matrices. I would like to find the solution of a linear system $Ax=B$ using the SVD decomposition of $A$ given by $A = U S V^t$. If I am not very wrong, I believe I can obtain this solution using


$\ \ \ X = VL^tU^tB$


in which


$\ \ \
 L\_{ij} =
 \begin{cases}
 1/S\_{ij} && \text{if} S\_{ij} \not= 0 \\\\
 0 && \text{otherwise}
 \end{cases}
$ 


This seems to work. However, for the case when $A$ has more columns than rows, I would like to find not only the best solution (in the least squares sense), but also the solution which has minimum norm. How can I do that?


$\\\\$


For example, by taking


$
\ \ \ A = \begin{pmatrix}
 1 & 2 & 3 \\\\
 4 & 5 & 6 
\end{pmatrix} \ \ \ \ \text{and} \ \ \ 
B = \begin{pmatrix}
 7 & 8 & 9 \\\\
 10 & 11 & 12 
\end{pmatrix}
$


The aforementioned method gives the solution


$\ \ \ \hat{X} = \begin{pmatrix}
 -2.1667 & -2.6667 & -3.1667 \\\\
 0.3333 & 0.3333 & 0.3333 \\\\
 2.8333 & 3.3333 & 3.8333 \\\\
\end{pmatrix}
$


I can verify this is a valid solution by taking 


$\ \ \ A\hat{X} = \begin{pmatrix}
 7 & 8 & 9 \\\\
 10 & 11 & 12 
\end{pmatrix}$


However, I am not sure this is the minimum norm solution.



Minimum norm solution of a least squares using SVD
Take a look at the wikipedia page on the [Moore-Penrose pseudoinverse](http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse#Minimum-norm_solution_to_a_linear_system), specifically Sections 5.3, 6.1, and 6.3.


