interpreting the multivariate Kalman filter update equations

Question

consider a multi-dimensional Kalman filter model with these state transition and measurement probabilities:

$P(x_{t+1} | x_{t}) = Normal(Fx_{t}, \Sigma_{x})$

$P(z_{t} | x_{t}) = Normal(Hx_{t}, \Sigma_{z})$

Following standard notation conventions the update equations for mean and covariance of the posterior over the next state $x_{t+1}$, $P(x_{t+1} | z_{t+1})$, are:

$\mu_{t+1} = F\mu_{t} + K_{t+1}(Z_{t+1} - HF\mu_{t})$

$\Sigma_{t+1} = (I - K_{t+1})(F\Sigma_{t}F^{T} + \Sigma_{x})$

where the Kalman gain at $t+1$ is $K_{t+1} = (F\Sigma_{t}F^{T} + \Sigma_{x})H^{T}(H(F\Sigma_{t}F^{T} + \Sigma_{x})H^{T} + \Sigma_{z})^{-1}$

Equation for $\mu_{t+1}$ makes sense. Questions:

1. In the equation for $\Sigma_{t+1}$, what does $F\Sigma_{t}F^{T}$ represent?
2. For Kalman gain: what is the intuition behind the equation, and why in the world is matrix inversion needed?
3. In Kalman gain calculation, what does the matrix inversion correspond to intuitively in the filtering step?

Answer 1

1. $F \Sigma_t F^T$ represents the covariance matrix of the state from the previous time step with a linear transform $F$ applied to it.

2 & 3. Matrix inversion comes out of the analytic representation of solutions to least squares problems. Consider $Ax=b$. A least squares solution to an overdetermined system is $x=(A^T A)^{-1} A^T b$.

Please see my post An Explanation of the Kalman Filter. I believe it will explain things a lot better.