Hessian matrix non-decomposition theorem (Q1574762)

In his invited lecture at the ICM Warsaw 1983 Roger Brockett proposed to classify finite-dimensional estimation algebras. The concept given independently by Brocket, Clark and Mitter plays an important role in the investigation of finite-dimensional nonlinear filters which are useful in both commercial and military industries. Yau's program is to show that Wong's antisymmetric matrix \(\Omega\) must have constant entries for a finite-dimensional estimation algebra. Together with a weak form of Hessian matrix nondecomposition theorem they solve the desired Brockett's problem of classification. The present paper deals with the question: Can the Hessian matrix of a homogeneous polynomial of degree 4 be decomposed into the form \(\Delta(x) \Delta(x)^T\) where \(\Delta(x)\) is an antisymmetric linear matrix? The answer is no! The antisymmetry of \(\Delta(x)\) is essential, an example illustrates this fact.