Hessian matrix non-decomposition theorem

In his 1983 invited lecture at the International Congress of Mathematics, Roger Brockett proposed to classify finite dimensional estimation algebras. The following problem arises from the first author’s classification theory of finite dimensional estimation algebras with maximal rank. Can the Hessian matrix of a homogeneous polynomial of degree 4 be decomposed in the form $\Delta(x)\Dta(x)^T$ where $\Delta(x)$ is an anti-symmetric linear matrix (i.e., entries of $\Delta(x)$ are linear in $x$)? In this short note, we show that this cannot be true, in other words, the Hessian matrix is nondecomposable in this form.

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Published 1 January 1999