# Proving that the uncertainty can not increase during the update step of a Kalman filter - positive semidefiniteness

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?

$$Q_t$$ is real-valued and positive definite, thus $$Q_t^{-1}$$ is real-valued and positive definite.

Now it's just making up a lemma of the Cholesky decomposition:

• If $$Q_t^{-1}$$ is real-valued and positive definite, then there's some real-valued $$q$$ such that $$q_t^T q_t = Q_t^{-1}$$ (Cholesky decomposition)
• If that holds, then $$\left(q_t C_t \right)^T\left(q_t C_t \right)$$ must be real-valued and at least positive semidefinite for any real-valued $$C_t$$ (Cholesky again -- that guy gets around)
• $$\left(q_t C_t \right)^T\left(q_t C_t \right) = C_t^T q_t^T q_t C_t = C_t^T Q_t^{-1} C_t$$
• Cogito, ergo sum. Or QED$$^*$$, or something brainy, and Latin.

$${\tiny ^* {\rm quod\ erat\ demonstrandum.}}$$

• BTW: I knew this in my bones, but I had to whomp up the proof -- thanks for the exercise! Nov 9, 2021 at 0:56
• I don't see the second point "$\left(q_t C_t \right)^T\left(q_t C_t \right)$ must be real-valued and at least positive semidefinite for any real-valued $C_t$" straightforward. Could you please elaborate? Thanks. Nov 9, 2021 at 9:44
• If there is some matrix $L$ with all real-valued entries, of any rank, then $L^T L$ is positive semidefinite. If $L$ is of rank $n$ and $L^T L$ is $n \times n$ then $L^T L$ is positive definite. So let $q_t C_t = L$. Nov 9, 2021 at 16:07