Proof that the Kalman gain minimizes the generalized variance

TL;DR

This paper proves that the Kalman gain not only minimizes the trace of the covariance matrix but also minimizes its determinant, known as the generalized variance, linking it to entropy minimization under Gaussian assumptions.

Contribution

The paper provides a novel proof that the Kalman gain minimizes the generalized variance, extending the understanding of its optimality beyond the trace minimization.

Findings

Kalman gain minimizes the determinant of the covariance matrix

Minimization of differential entropy under Gaussian errors

Extension of optimality criteria for Kalman filter

Abstract

The optimal gain matrix of the Kalman filter is often derived by minimizing the trace of the posterior covariance matrix. Here, I show that the Kalman gain also minimizes the determinant of the covariance matrix, a quantity known as the generalized variance. When the error distributions are Gaussian, the differential entropy is also minimized.