In Kalman filtering, you start with an initial estimate $P_0$, then refine that estimate at each step with $$P_{n | n} = \left( I - K_n H_n \right)P_{n | n-1} \tag{1}$$ where $H_n$ is known a-priori, and $$K_n = P_{n | n-1} H_n^T\left(H_n P_{n | n-1} H_n^T + R_n\right)^{-1} \tag 2$$ where, again, $R_n$ is known a-priori.
$P_n$ is a square, positive-definite matrix. In general, $H_n$ is a $m \times n$ matrix.
In some problems, you have prior knowledge of the likely starting value of your estimate, so $P_0$ is finite. Sometimes, it is convenient to model your system as having no clue whatsoever of the starting values -- in this case $P_0$ would have infinite eigenvalues. I'm pretty sure that, without loss of generality, you could say that $P_0 = \infty I$.
In doing so, one immediately runs into problems with a mix of infinities and real values. I know both from 1-dimensional examples and from intuition into the workings of the Kalman filter, that as soon as the first update to the covariance matrix (1) is undertaken, $P_{1|1}$ should have $m$ finite eigenvalues. Moreover, in a well-formed problem (and again, intuitively), at each measurement $P{n|n}$ gains finite eigenvalues and loses infinite ones until eventually it's altogether finite. But I'd like to know if there's a way to actually do the linear algebra/calculus to show this.
If you crank through the one-dimensional case, it's basically the matrix equivalent of $$p_{1|1} = \lim_{p_{0|0} \to \infty} p\left(\frac{1}{p_{0|0}} - \frac{h^2}{p_{0|0} h^2 + r}\right )$$
So -- how can I go from my one-dimensional case to the multidimensional case? Is this a solved problem? If answers come with search terms or subject areas to go buy books under, that would be even more appreciated.