Mathematical Research Letters

Volume 6, Issue 6, November 1999 pp. 663-673.

Hessian matrix non-decomposition theorem

Authors: Stephen S.-T. Yau, Wing-Shing Wong, and Xi Wu
Author institution: University of Illinois at Chicago, and The Chinese University of Hong Kong

Summary: 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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