Sum of squares representation for the B\"ottcher-Wenzel biquadratic form

TL;DR

This paper determines the conditions under which the Böttcher-Wenzel biquadratic form can be expressed as a sum of squares, providing solutions for certain matrix classes and conjecturing for others.

Contribution

It introduces the minimal scale factor for sos representation of the Böttcher-Wenzel form and provides primal and dual solutions for the related semidefinite program.

Findings

The form is sos for tridiagonal, backward tridiagonal, and cyclic Hankel matrices.

Primal and dual solutions for the semidefinite program are provided.

Conjecture on sos representability for Toeplitz matrices.

Abstract

We find the minimum scale factor, for which the nonnegative B\"ottcher-Wenzel biquadratic form becomes a sum of squares (sos). To this we give the primal and dual solutions for the underlying semide finite program. Moreover, for special matrix classes (tridiagonal, backward tridiagonal and cyclic Hankel matrices) we show that the above form is sos. Finally, we conjecture sos representability for Toeplitz matrices.