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Given , a system

might be over- or under-determined. We still want to compute an approximate solution.

For any such , there exists the Singular Value Decomposition
where
is orthonormal:

is orthonormal:

From this, we can construct the pseudo-inverse

where
where

Due to numerical imprecisions, we use


Now, we can compute a solution:





This solution is

• a solution in the least-squares sense if M is overdetermined
• a solution in the last-norm sense if M in underdetermined