About the generalized \(LM\)-inverse and the weighted Moore-Penrose inverse. (English) Zbl 1191.65038
The weighted Moore-Penrose inverse \(X\) of the matrix \(A\in{\mathbb C}^{m\times n}\) is defined by the equations \(AXA=A\), \(XAX=X\), \((MAX)^*=MAX\), \((NXA)^*=NXA\) where \(M\) and \(N\) are positive definite weighting matrices and the superscript * denotes the conjugate transpose. If the weighting matrices are unit matrices, this gives the ordinary Moore-Penrose inverse. An algorithm to compute this weighted Moore-Penrose inverse was given by G. Wang and Y. Chen [J. Comput. Math. 4, 74–85 (1986; Zbl 0617.65033)].
The generalized \(LM\)-inverse of \(A\) is defined as the matrix \(A^+\) such that \(x=A^+b\) minimizes \(\|L^{1/2}(Ax-b)\|^2=\|Ax-b\|_L\) and \(\|M^{1/2}x\|=\|x\|_M\). Again \(L\) and \(M\) are positive definite weighting matrices. An algorithm to compute the this generalized inverse was given by F. E. Udwadia and P. Phohomsiri [Appl. Math. Comput. 190, No. 2, 999–1006 (2007; Zbl 1125.15007)].
Both algorithms are explicitly given and then it is proved that both algorithms are equivalent and compute the same result. The appendix of this paper gives the Mathematica code of the algorithms.
The generalized \(LM\)-inverse of \(A\) is defined as the matrix \(A^+\) such that \(x=A^+b\) minimizes \(\|L^{1/2}(Ax-b)\|^2=\|Ax-b\|_L\) and \(\|M^{1/2}x\|=\|x\|_M\). Again \(L\) and \(M\) are positive definite weighting matrices. An algorithm to compute the this generalized inverse was given by F. E. Udwadia and P. Phohomsiri [Appl. Math. Comput. 190, No. 2, 999–1006 (2007; Zbl 1125.15007)].
Both algorithms are explicitly given and then it is proved that both algorithms are equivalent and compute the same result. The appendix of this paper gives the Mathematica code of the algorithms.
Reviewer: Adhemar Bultheel (Leuven)
MSC:
| 65F20 | Numerical solutions to overdetermined systems, pseudoinverses |
| 15A09 | Theory of matrix inversion and generalized inverses |
Keywords:
generalized \(LM\)-inverse; weighted Moore-Penrose inverse; rational matrices; MATHEMATICA; partitioning methodSoftware:
MathematicaReferences:
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