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, 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

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

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

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