Positive definite matrix and pseudoinverse

Question

I have a matrix $J \in R^{2p \times n}$ with $2p>n$ and I know that $rank(J)=n$. Is $J^+ J$ a positive definite matrix? If not, are there any assumptions I can make in order for it to be positive definite?

Answer 1

$\operatorname{Rank}(J)=n$ means that the columns of $J$ are linearly independent and you have $J^+ = (J^T J)^{-1}J^T$, i.e. $J^+ J = I_n$ which is indeed positive definite.