Let $\displaystyle P^ + ,P^ - ,I,Q \in R^{n\times n}, K\in R^{n\times 1}, M\in R^{1 \times n}$, and assume that $\displaystyle Q$ is positive definite, $\displaystyle P^ -$ is positive semidefinite whence $\displaystyle (MP^ - M^T + Q)^{ - 1}$ exists (where $\displaystyle T$ denotes transpose).

In what sense does $\displaystyle K = P^ - M^T(MP^ - M^T + Q)^{ - 1}$ minimize the quadratic expression $\displaystyle P^ + : = (I - KM)P^ - (I - KM)^T + KQK^T$, over $\displaystyle K$?

Is this minimization of $\displaystyle P^ +$ over all vectors $\displaystyle K$ with respect to the usual ordering for positive semidefinite matrices $\displaystyle A\le B$ iff $\displaystyle B - A $is positive semidefinite?

Next consider the extension $\displaystyle P^ + : = (I - KAM)P^ - (I - KAM)^T + KAQA^TK^T$, where $\displaystyle A\in R^{n\times 1}, K\in R^{n\times n}$ and $\displaystyle K$ diagonal, where all other matrices are as above.

What is the minimum over $\displaystyle K$ (with respect to the previous ordering or something )??

Any help will be deeply appreciated.