I am trying to prove mathematically that the update step in a Kalman filter can not result in a increase in uncertainty. I found the following proof which is based on the inversion lemma and the concept of positive semidefiniteness, but I can't grasp why the matrix product in the end is positive semi-definite.
Proof:
The uncertainty in the belief after the update step is expressed by the covariance matrix :
$\Sigma_t = (I - K_tC_t)\bar{\Sigma_{t}}$ (1)
where $\bar{\Sigma_t}$ is the predicted covariance before the update and $K_t$ is the Kalman Gain term, which is defined as:
$K_t = \bar{\Sigma_t}{C_t}^T(C_t\bar{\Sigma_t}{C_t}^T + {Q_t})^{-1}$ (2)
where $C_t$ and $Q_t$ are the parameters of the $z_t$ measurement's distribution $p(z_t | x_t) \sim \mathcal{N}(C_tx_t, Q_t) $
By expanding the $K_t$ term in equation (1) and then by applying the matrix inversion lemma/Woodburry matrix identity, we get:
$ \Sigma_t = \bar{\Sigma_{t}} - \bar{\Sigma_t}{C_t}^T(C_t\bar{\Sigma_t}{C_t}^T + {Q_t})^{-1}C_t\bar{\Sigma_t} \\ = (\bar{\Sigma}^{-1}_t + {C_t}^T{Q^{-1}_t}C_t)^{-1}$
The proof mentions that ${C_t}^T{Q^{-1}_t}C_t$ is positive semi-definite and hence the uncertainty cannot increase.
End of proof
Question: I know that $Q_t$ is positive semi-definite since it is a covariance matrix but I haven't been able to understand why ${C_t}^T{Q^{-1}_t}C_t$ is positive semi-definite. Can someone explain?