The pseudoinverse is an extension of the idea of the inverse of a nonsingular sq. matrix to singular matrices and rectangular matrices. It’s one in all many generalized inverses, however the one most helpful in apply because it has plenty of particular properties.
The pseudoinverse of a matrix 
Right here, the superscript 
An important property of the pseudoinverse is that for any system of linear equations 
The pseudoinverse might be expressed by way of the singular worth decomposition (SVD). If 
the place the diagonal matrix is 
pinv computes
From the Moore–Penrose situations or (1) it may be proven that 
For full rank 
Consequently,
Some particular instances are price noting.
The pseudoinverse differs from the standard inverse in numerous respects. For instance, the pseudoinverse of a triangular matrix is just not essentially triangular (right here we’re utilizing MATLAB with the Symbolic Math Toolbox):
>> A = sym([1 1 1; 0 0 1; 0 0 1]), X = pinv(A) A = [1, 1, 1] [0, 0, 1] [0, 0, 1] X = [1/2, -1/4, -1/4] [1/2, -1/4, -1/4] [ 0, 1/2, 1/2]
Moreover, it’s not typically true that 
It’s not normally essential to compute the pseudoinverse, however whether it is required it may be obtained utilizing (1) or (2) or from the Newton–Schulz iteration
for which 
Notes and References
The pseudoinverse was first launched by Eliakim Moore in 1920 and was independently found by Roger Penrose in 1955. For extra on the pseudoinverse see, for instance, Ben-Israel and Greville (2003) or Campbell and Meyer (2009). For evaluation of the Newton–Schulz iteration see Pan and Schreiber (1991).
- Adi Ben-Israel and Thomas N. E. Greville, Generalized Inverses: Concept and Purposes, second version, Springer-Verlag, New York, 2003
- Stephen Campbell and Carl Meyer, Generalized Inverses of Linear Transformations, Society for Industrial and Utilized Arithmetic, Philadelphia, PA, USA, 2009. printed (Initially printed by Pitman in 1979.)
- Victor Pan and Robert Schreiber, An Improved Newton Iteration for the Generalized Inverse of a Matrix, with Purposes, SIAM J. Sci. Statist. Comput. 12 (5), 1109–1130, 1991.
