EKF update with 0 uncertainty measurement noise

Question

I was trying to understand how the Kalman gain works, from what I understand it determines the reliance on the measurement based on current state and measurement uncertainty.

The Kalman gain, K, is given by

$$ K_k = P_k H^T (H P_k H^T + R)^{-1} $$

for predicted state covariance $P_k$, measurement Jacobian $H$, and measurement noise $R$. I am using values from an example discussed in this other question.

$$ P_k = \left( \begin{array}{cc} 200.25 & 100.5 \\ 100.5 & 101 \end{array} \right) $$

$$ H = \left( \begin{array}{cc} 1 & 0 \end{array} \right) $$

so for $R = \left( 0 \right)$, I am getting $$K_k = \left(\begin{array}{c} 1.0000 \\ 0.5019 \end{array} \right).$$

However simplification of the Kalman gain equation for $R = \left( 0 \right)$ becomes $$ K_k = P_k H^T (H P_k H^T + 0)^{-1} $$ $$ K_k = P_k H^T (H P_k H^T)^{-1} $$ $$ K_k = P_k H^T (P_k H^T)^{-1} H^{-1} $$ $$K_k = H^{-1}$$ Which in my opinion is a pseudo inverse, accordingly $$K_k = \left(\begin{array}{c} 1.0000 \\ 0.0 \end{array} \right).$$

am I violating anything fundamental, or is it just computer approximation error?

Answer 1

The simplification of the Kalman gain attempted in the question is mistaken. $H$ and $P_k\,H^\mathrm{T}$ are not square matrices in this case (the measurement and the state have different dimensions) and thus $(H \, P_k \, H^\mathrm{T})^{-1}$ cannot be written as $(P_k\, H^\mathrm{T})^{-1}\,H^{-1}$. The latter expression does not even make sense since $H$ and $P_k\,H^\mathrm{T}$ do not have inverses.

(Pseudoinverse exists, but it does not have all properties inverse has.)