  The Kalman gain is commonly derived as the minimizer of the trace of
theposterior covariance. It is known that it also minimizes the determinant of
the posterior covariance. I will show that it also minimizes the smallest
Eigenvalue $\lambda_1$ and the chracteristic polynomial on
$(-\infty,\lambda_1)$ and is critical point to all symmetric polynomials of the
Eigenvalues, minimizing some. This expands the range of uncertainty measures
for which the Kalman Filter is optimal.
