In his 1983 invited lecture at the International Congress of Mathematics, R
oger 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 Hessia
n matrix of a homogeneous polynomial of degree 4 be decomposed in the form
Delta(x)Delta(z)(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 th
is cannot be true, in other words, the Hessian matrix is nondecomposable in
this form.