MPINVERT: Moore-Penrose Inverse of a Matrix
02-06-2015, 07:38 PM (This post was last modified: 02-06-2015 07:49 PM by Han.)
Post: #4
 Han Senior Member Posts: 1,882 Joined: Dec 2013
RE: MPINVERT: Moore-Penrose Inverse of a Matrix
The SVD decomposition of a matrix can also be used to compute the pseudo-inverse of a matrix. If
$A = U \cdot \Sigma \cdot V^T$
where $$\sigma_i$$ are the singular values of the diagonal matrix $$\Sigma$$, and $$U$$ and $$V$$ are unitary matrices.

The pseudo-inverse of $$A$$ is
$A^{+} = V \cdot \Sigma^{+} \cdot U^T$
where $$\Sigma^{+}$$ is obtained by replacing the diagonal entries $$\sigma_i$$ with $$1/\sigma_i$$ for $$\sigma_i > \epsilon$$ and 0 otherwise and transposing. If $$A$$ is non-singular, then all $$\sigma_i$$ are non-zero and the pseudo-inverse is the same as the regular inverse. On the other hand, if $$A$$ is singular, then only the "non-zero" (greater than $$\epsilon$$) singular values are inverted in computing $$A^{+}$$.

It's a bit more powerful in that it even handles cases when $$A^TA$$ or $$AA^T$$ are not invertible. The SVD has lots of neat applications! What are they? Well, that's a google exercise left to the diligent reader :-)

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 Messages In This Thread MPINVERT: Moore-Penrose Inverse of a Matrix - Eddie W. Shore - 08-29-2014, 09:30 PM RE: MPINVERT: Moore-Penrose Inverse of a Matrix - Namir - 09-01-2014, 09:11 PM RE: MPINVERT: Moore-Penrose Inverse of a Matrix - Han - 02-06-2015, 07:48 PM RE: MPINVERT: Moore-Penrose Inverse of a Matrix - salvomic - 02-06-2015, 03:09 PM RE: MPINVERT: Moore-Penrose Inverse of a Matrix - Han - 02-06-2015 07:38 PM RE: MPINVERT: Moore-Penrose Inverse of a Matrix - salvomic - 02-06-2015, 08:40 PM RE: MPINVERT: Moore-Penrose Inverse of a Matrix - Han - 10-27-2015, 06:57 PM RE: MPINVERT: Moore-Penrose Inverse of a Matrix - salvomic - 10-28-2015, 09:02 AM

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