Let $P^ + ,P^ - ,I,Q \in R^{n\times n}, K\in R^{n\times 1}, M\in R^{1 \times n}$, and assume that $Q$ is positive definite, $P^ -$ is positive semidefinite whence $(MP^ - M^T + Q)^{ - 1}$ exists (where $T$ denotes transpose).
In what sense does $K = P^ - M^T(MP^ - M^T + Q)^{ - 1}$ minimize the quadratic expression $P^ + : = (I - KM)P^ - (I - KM)^T + KQK^T$, over $K$?
Is this minimization of $P^ +$ over all vectors $K$ with respect to the usual ordering for positive semidefinite matrices $A\le B$ iff $B - A$is positive semidefinite?
Next consider the extension $P^ + : = (I - KAM)P^ - (I - KAM)^T + KAQA^TK^T$, where $A\in R^{n\times 1}, K\in R^{n\times n}$ and $K$ diagonal, where all other matrices are as above.
What is the minimum over $K$ (with respect to the previous ordering or something )??