Defining the right generalized inverse of a non-square Jacobian matrix $J$, $J^{\#}$, as


$J^{\#} = M^{-1} J^T \left(J M^{-1} J^T\right)^{-1}$


where the matrix $M \succ 0$ is positive definite and symmetric, can we infer that the following null space projection matrix


$\left(I - J^\# J \right)$


is non-negative definite?


For the engineering problem that I am tackling, I was able to show that


$M\left( I - J^\# J \right) \succeq 0$



Null space vs. semi-positive definite matrix
No, consider the following counterexample: Take $$ M = \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix} , \quad J = \begin{pmatrix} 1 & 2\end{pmatrix},$$ then $J^\# = \begin{pmatrix} 1 \\ 0\end{pmatrix}$ and your projection is given by $I - J^\# J = \begin{pmatrix} 0 & -2\\ 0 & 1\end{pmatrix}$ and this is definitely not non-negative definite by your definition. 


Edit: Can't seem to get the matrices right...


