I'm working through the derivation for the Kalman filter using orthogonal projections. It's pretty intuitive that if we want to generate the best linear estimate for an $n$-vector $x$ using some $m$-vectors $y_0, \dots, y_t$, we would take the orthogonal projection of $x$ onto the subspace $\mathbb{R}^m$ spanned by the vectors $y_i$.
In the Kalman filter case, it seems that we assume $n>m>t$. That is, a lower dimensional observations $(y)$ than states $(x)$ we are trying to estimate. Similarly, we are trying to build a recursive estimator that uses additional observations $y_i$ such that at timestep $t$ we have $t$ observation vectors and our subspace is of dimension $t$.
My question is this: what happens when $t=m=n$? This would mean, I think, that the orthogonal projection is meaningless and it wouldn't help your filter to gain new estimates... is this correct?
