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.
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