Kalman filter innovation residual inversion I'm trying to implement a Kalman filter in a computationally efficient way. The main issue is the inversion of the innovation residual: $$S=HPH^T+R$$ $$K=PH^TS^{-1}$$ My question is, can one assume that the S matrix in positive definite? This would make inverting it more computationally efficient....

If $P$ is non negative-definite and $R$ is positive-definite, then yes $S$ is positive definite. $S$ is positive definite if for every non-zero $x$, \begin{align} 0 &< x^T S x \\\ 0 &< x^T ( HPH^T + R ) x \\\ 0 &< x^THPH^Tx + x^T R x \\\ 0 &< (H^Tx)^TP(H^Tx) + x^T R x \end{align} If $R$ is positive definite then $ x^T R x > 0 $. If $P$ is non-negative definite then $(H^Tx)^TP(H^Tx) \ge 0$. If $R$ is positive definite and $P$ is non-negative definite then $(H^Tx)^TP(H^Tx) + x^T R x > 0$ and so $S$ is positive definite.

Generally covariance matricies ($P$, $R$) are positive-definite so generally $HPH^T+R$ is positive definite.