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.

翻译：卡尔曼增益通常被推导为后验协方差迹的最小化器。已知它也能最小化后验协方差的行列式。我将证明它还能最小化最小特征值$\lambda_1$及其在$(-\infty,\lambda_1)$上的特征多项式，并且是特征值所有对称多项式的驻点，同时使其中一些达到极小值。这扩展了卡尔曼滤波器最优时所适用的不确定性度量范围。