I am currently in the process of learning Kalman Filters and facing the following Equations: $$\vec x_k = F\vec x_{k-1} + B\vec u_k$$ $$P_k = FP_{k-1}F^T + Q$$
$\vec x_k$: state vector.
$P_k$: Covariance matrix.
$F$: state space transformation matrix.
$B$: state space transformation matrix for external forces.
$Q$: noise matrix.
My Question is about the $FP_{k-1}F^T$ part. I've seen this pattern through out my engineering studies, but I don't know what it actually means to left and right multiply something by the same matrix.
It's basically a change of basis for matrix $P_{k-1}$. Suposse we a rotation matrix $A$ which can change the components of a vector from a orthonormal basis $B_{old}$ to a basis $B_{new}$ as follows: $$ [v_{new}]_{n \times 1}=[A]_{n \times n}[v_{old}]_{n \times 1} \quad \textbf{[1]} $$
Because $A$ is orthonormal, then $A^{-1}=A^T$.
Suposse we have some particular transformation of a vector $\omega$ in basis $B_{old}$ associated with a matrix $M$, such as: $$ [H]_{n \times 1}=[M]_{n \times n}[\omega]_{n \times 1} $$
But maybe, we don't want vector $H$ in terms of basis $B_{old}$ but in terms of $B_{new}$, so, using equation $\textbf{[1]}$: $$ H_{old}=M\omega_{old} $$
$$ A^{T}H_{new}=MA^{T}\omega_{new} $$
$$ H_{new}=(AMA^{T})\omega_{new} $$
For convenience, let's call $M=M_{old}$. Now, the new transformation matrix $M_{new}$ in basis $B_{new}$ would be: $$ M_{new}=AM_{old}A^{T} $$
IMPORTANT NOTE: In some textbooks you could find another definition:
$$ M_{new}=A^{T}M_{old}A $$
This one comes from the fact that they assume matrix $A$ can transform from basis $B_{new}$ to $B_{old}$ like this: $$ [v_{old}]_{n \times 1}=[A]_{n \times n}[v_{new}]_{n \times 1} $$
So, you should always have clear this fact in your mind to know which definition you have to use.
Regards.