MPINVERT: MoorePenrose Inverse of a Matrix

08292014, 09:30 PM
Post: #1




MPINVERT: MoorePenrose Inverse of a Matrix
Also known as a pseudoinverse, the MoorePenrose inverse of a matrix, denoted by A^+ (capital A with a supersubscript of a plus sign), is an inverse of matrix. Different from the "true" matrix inverse, the MoorePenrose inverse allows for nonsquare matrices. Primarily, the MoorePenrose inverses are calculated is assist in solving linear leastsquare equations.
Input: MPINVERT(matrix). Execute this from Home or CAS screen. Program: EXPORT MPINVERT(mtx) BEGIN // 20140827 EWS // MoorePenrose Matrix Inverse LOCAL r,c,d,n; d:=SIZE(mtx); r:=d(1); c:=d(2); n:=RANK(mtx); CASE IF n==c THEN RETURN (TRN(mtx)*mtx)^1*mtx; END; IF n==r THEN RETURN TRN(mtx)*(mtx*TRN(mtx))^1; END; DEFAULT RETURN "No Solution Found"; END; END; Examples Matrix: [ [ 1, 2, 3 ] [ 3, 4, 0 ] ] MoorePenrose Inverse: [ [ 8/229, 31/229 ] [ 6/229, 34/229 ] [ 75/229, 33/229 ] ] Matrix: [ [7, 4, 6, 7] [1, 5, 3, 3] ] MoorePenrose Inverse: (to four decimal places) [ [0.0489, 0.0338] [0.0194, 0.1092] [0.0360, 0.0600] [0.0520, 0.0800] ] 

09012014, 09:11 PM
Post: #2




RE: MPINVERT: MoorePenrose Inverse of a Matrix
In the case of square matrices, how close should the PM pseudoinverse be to the actual matrix inverse?
Namir 

02062015, 03:09 PM
Post: #3




RE: MPINVERT: MoorePenrose Inverse of a Matrix
(08292014 09:30 PM)Eddie W. Shore Wrote: Also known as a pseudoinverse, the MoorePenrose inverse of a matrix, denoted by A^+ (capital A with a supersubscript of a plus sign), is an inverse of matrix. ... hi Eddie, thank you! very useful for me. However I think that the "left" pseudo inverse should me Code:
also you could use Code:
without calculate at all d:=SIZE(mtx) and so on... Obviously you should set also Local r,c,n; I hope you could appreciate my little two cents Salvo ∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C  DM42, DM41X  WP34s Prime Soft. Lib 

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 

02062015, 07:48 PM
Post: #5




RE: MPINVERT: MoorePenrose Inverse of a Matrix
(09012014 09:11 PM)Namir Wrote: In the case of square matrices, how close should the PM pseudoinverse be to the actual matrix inverse? If A is square, and A is invertible, then they should be exactly equal. If A is invertible, then so is A^T and \[ A^{+} = (A^TA)^{1} A^{T} = A^{1} (A^T)^{1} A^T = A^{1} \] Graph 3D  QPI  SolveSys 

02062015, 08:40 PM
(This post was last modified: 02062015 09:07 PM by salvomic.)
Post: #6




RE: MPINVERT: MoorePenrose Inverse of a Matrix
(02062015 07:38 PM)Han Wrote: The SVD decomposition of a matrix can also be used to compute the pseudoinverse of a matrix. If Hi Han, I'm interested in this extension, almost for the case in witch the matrix is not invertible and MPINVERT doesn't work. I saw the SVD syntax. The guide writes "Description Singular Value Decomposition. Factors an m n matrix into two matrices and a vector: {[[m m square orthogonal]],[[n n square orthogonal]], [real]}.". Now I try svd([[1,2][1,2]]) and I get: a matrix [[1/√2,0][1/√2,0]] then a vector [3.16227766..., 0] and at the end another matrix [[0.447213..., 0.894427...][0.894427..., 0.4472135955]] What it the "real"? Those values are U, ∑, V and ∑ is the diagonal (√10, 0). Is it right? And therefore, how can we operate to get ∑, A+ and so on? Eddie, then, could improve it's program for the case of singular non invertible matrix... Another thing: in that case Prime apparently give a vector with 3 elements (matrix, vector, matrix): there is a simple way to extract one of them, i.e. to make ration approximation (with QPI)? Please, help. Salvo ∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C  DM42, DM41X  WP34s Prime Soft. Lib 

10272015, 06:57 PM
(This post was last modified: 10272015 07:01 PM by Han.)
Post: #7




RE: MPINVERT: MoorePenrose Inverse of a Matrix
(02062015 08:40 PM)salvomic Wrote: Hi Han, The vector of "real" values is a list of the singular values (i.e. the diagonal entries of \( \Sigma \). Quote:Those values are U, ∑, V and ∑ is the diagonal (√10, 0). Is it right? Yes. The help has been updated to correct the ordering of the output so that now it does in fact return \( \{ U, \Sigma, V \} \). Quote:And therefore, how can we operate to get ∑, A+ and so on? The pseudo inverse is \( V \Sigma^{1} U^{H} \) where \( U^H \) is the Hermitian (conjugate transpose for complex matrices, or just transpose for realvalued matriced) and \( \Sigma^{1} \) is simply the the transpose of \( \Sigma \) with the diagonal entry \( s_i \) replaced by \( \frac{1}{s_i} \). For values of \( s_i \) that are very small, (i.e. 0), then \( s_i \) is simply replaced by 0. Quote:Another thing: in that case Prime apparently give a vector with 3 elements (matrix, vector, matrix): there is a simple way to extract one of them, i.e. to make ration approximation (with QPI)? Please, help. You can use mat2list, and QPI accepts lists as input. Currently, the SVD() command is not as robust. I recommend using: http://hpmuseum.org/forum/thread4976.html I have updated the program to also include pinv() for pseudoinverse as well as pivoted QR factorization. This article has a decent interpretation of the SVD (though nowhere complete, however): http://robotics.caltech.edu/~jwb/courses...pseudo.pdf Graph 3D  QPI  SolveSys 

10282015, 09:02 AM
Post: #8




RE: MPINVERT: MoorePenrose Inverse of a Matrix
thank you, Han,
I'll try soon your new program! many thanks for the explanation of SVD syntax Salvo ∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C  DM42, DM41X  WP34s Prime Soft. Lib 

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