Positive Definiteness of a symmetrized matrix involving the Moore Penrose Pseudoinverse

Question

Let $M\in\mathbb{R}^{m\times n }$ and $V\in\mathbb{R}^{n\times l }$ both have full column rank and let $X^{+}$ denote the Moore Penrose Pseudoinverse of the matrix $X$.

Question: Is the symmetrized matrix $V[M^{\top}MV]^{+}+(V[M^{\top}MV]^{+})^{\top}\in\mathbb{R}^{n\times n }$ positive(semi)-definite?

Answer 1

The matrix $V[M^{\top}MV]^{+}$ cannot have negative eigenvalues, however, i found examples where $V[M^{\top}MV]^{+} + (V[M^{\top}MV]^{+})^{\top}$ has negative eigenvalues, so the answer is in general no.