Singular value decomposition and inverse of square matrix

Question

I've previously touched the subject in this question. There (and subsequently on other places), I've learned that if a SVD is applied to a square matrix $M$, $M=USV^T$, then the inverse of $M$ is relatively easy to calculate as $M^{-1}=V S^{-1}U^T$. I've implemented the SVD algorithm and began to receive wrong results, so I fed my test examples to Matlab and was surprised to find that $M^{-1}=V S^{-1}U^T$ apparently doesn't hold.

So, my question is am I calculating the inverse of a matrix based on its SVD correctly? Am I missing something there?

Please note that I'm not asking for debugging help, seeking bugs in Matlab, etc. I'm just perplexed that the equation $M^{-1}=V S^{-1}U^T$ doesn't hold for some reason, as, when calculated separately, $M^{-1}$ and $V S^{-1}U^T$ give different results.

Here is an example (the results come from Matlab and have nothing to do with any implementation of mine):

 M = 32.7276   -5.0470   -5.3461   -1.7619
     -5.0470   10.1665   -5.1195   -2.0058
     -5.3461   -5.1195   38.7891   10.4173
      1.7619    2.0058  -10.6087   38.5192

$M=USV^T$ (obtained with the command [U S V] = svd(M))

U = -0.4313   -0.0317    0.8703    0.2355
    -0.0785   -0.0293   -0.2974    0.9511
     0.8860    0.1502    0.3905    0.1999
    -0.1509    0.9877   -0.0402    0.0054

S = 43.3263         0         0         0
          0   39.9753         0         0
          0         0   31.9654         0
          0         0         0    7.8516

V = -0.4321    0.0012    0.8705    0.2356
    -0.0799    0.0269   -0.2971    0.9511
     0.8927   -0.1084    0.3893    0.1996
     0.1001    0.9937    0.0495   -0.0042

Now, $V S^{-1}U^T$ (obtained with the command V.' * inv(S) * U.', and I am aware that this is a non-conjugate transpose, however the case is a real matrix) yields:

0.0317    0.0047    0.0043   -0.0015
0.0268    0.1214    0.0241    0.0015
0.0037    0.0010    0.0227   -0.0108
0.0022   -0.0035    0.0107    0.0224

While a direct inverse of $M$ (command inv(M)) yields:

 0.0351    0.0212    0.0078    0.0006
 0.0212    0.1181    0.0191    0.0020
 0.0078    0.0190    0.0277   -0.0061
-0.0006   -0.0019    0.0063    0.0241

The two should be the same, but clearly are not.

Answer 1

Now, $VS^{−1}U^T$ (obtained with the command U.' * inv(S) * V.'

This is wrong, the command should be V*inv(S)*U', which yields the answer you are looking for.

>> [U,S,V] = svd(M)
   inv(M) - V*inv(S)*U'

   ans =

   1.0e-15 *

   0.0625    0.0243   -0.0954    0.0181
   0.0243    0.1388    0.0451   -0.0017
   0.0408    0.0486   -0.0278    0.0052
   0.0035   -0.0004   -0.0026         0