MPINVERT: MoorePenrose Inverse of a Matrix

02062015, 07:38 PM
(This post was last modified: 02062015 07:49 PM by Han.)
Post: #4




RE: MPINVERT: MoorePenrose Inverse of a Matrix
The SVD decomposition of a matrix can also be used to compute the pseudoinverse 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 pseudoinverse 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 nonsingular, then all \( \sigma_i \) are nonzero and the pseudoinverse is the same as the regular inverse. On the other hand, if \( A \) is singular, then only the "nonzero" (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 :) Graph 3D  QPI  SolveSys 

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Messages In This Thread 
MPINVERT: MoorePenrose Inverse of a Matrix  Eddie W. Shore  08292014, 09:30 PM
RE: MPINVERT: MoorePenrose Inverse of a Matrix  Namir  09012014, 09:11 PM
RE: MPINVERT: MoorePenrose Inverse of a Matrix  Han  02062015, 07:48 PM
RE: MPINVERT: MoorePenrose Inverse of a Matrix  salvomic  02062015, 03:09 PM
RE: MPINVERT: MoorePenrose Inverse of a Matrix  Han  02062015 07:38 PM
RE: MPINVERT: MoorePenrose Inverse of a Matrix  salvomic  02062015, 08:40 PM
RE: MPINVERT: MoorePenrose Inverse of a Matrix  Han  10272015, 06:57 PM
RE: MPINVERT: MoorePenrose Inverse of a Matrix  salvomic  10282015, 09:02 AM

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