# If a matrix is positive semi-definite is the associated jacobian also positive semi-definite?

I'm studying the consistency of the Extended Kalman Filter and I'm trying to work through a proof that the covariance matrix is monotonically decreasing.

Theorem 1 from this paper, shows the proof for the Kalman Filter, so I'm working along the same lines.

The proof for the KF uses the fact that the process and observation matrices are positive semi-definite. For the EKF, the proof relies on the jacobians of the process and observation functions.

My question is: are these jacobian matrices also positive semi-definite, and why?

I believe this is true, but I'm not sure how to approach it. I know a matrix $A$ is positive semi-definite if its eigenvalues are all greater than 0. I don't know where to start to work out is if this property holds for the jacobian as well.