Hessian matrix non-decomposition theorem and its application to nonlinear filtering

Abstract

In 1983, Roger Brockett gave his invited lecture at the International Congress of Mathematics about the classification of finite dimensional estimation algebra (FDEA), which plays a vitally important role in the classification of finite dimensional filters (FDFs). The Hessian matrix non-decomposition (HMND) theorem was one of the key steps for the complete classification of FDEAs with maximal rank of the arbitrary state space dimension (Yau, 2003; Yau and Hu, 2005). In (Yau et al., 1999), the HMND theorem for homogeneous polynomials of degree 4 was proven and at the same time, it is firstly proposed that there exists a HMND conjecture for the general polynomials of degree 2d with d≥2. In this paper, we consider the Hessian matrix of a nontrivial real homogeneous polynomial at degree 6. For the state space dimension n≤4, we prove such matrix cannot be decomposed in the form Q(x)Q(x)T, where Q(x) is a skew-symmetric matrix whose all elements are homogeneous polynomials of degree 2, and Q(x)T is the transpose of Q(x). The fundamental strategy we use in this paper is that we generalized the techniques proposed by Yau et al. (1999) for the case of degree 4 and derived identities of coefficient matrices for the case of degree 6. For the arbitrary state space dimension n, the HMND theorem is demonstrated under several conditions by certain basic linear algebra techniques. These results can be closely related to the classification of FDEAs with non-maximal rank in the filtering theory.