# Moore--Penrose Inverse

For an arbitrary

*m*×*n*matrix**A**and self-adjoint (Hermitian) positive definite matrices**M**and**N**of order*m*and*n*, respectively, there is a unique matrix*n*×*m*matrix**G**satisfying the following equations:
\[
{\bf A}\,{\bf G}\,{\bf A} = {\bf A}, \quad {\bf G}\,{\bf A}\,{\bf G} = {\bf G} , \quad \left( {\bf M}\,{\bf A}\,{\bf G} \right)^{\ast} = {\bf M}\,{\bf A}\,{\bf G} , \quad \left( {\bf N}\,{\bf G}\,{\bf A} \right)^{\ast} = {\bf N}\,{\bf G}\,{\bf N} .
\]

Matrix **G**is known as the**weighted Moore–Penrose inverse**of**A**and is denoted by**A**^{†}. In particular, when*m*×*m*matrix**M**is the identity matrix and*n*×*n*matrix**N**is the identity matrix, the matrix**G**that satisfies the above conditions is recognized as the Moore–Penrose inverse or pseudoinverse.-
Wei, Y. and Wang, D., Condition numbers and perturbation of the weighted Moore–Penrose inverse and weighted linear least squares problem,
*Applied Mathematics and Computation*, Volume 145, Issue 1, 20 December 2003, Pages 45-58; https://doi.org/10.1016/S0096-3003(02)00437-X