On matrix algebras associated to sum-of-squares semidefinite programs

摘要

To each semidefinite program (SDP) in primal form, we associate the matrix algebra generated by its constraint matrices. In this note, we show that this algebra is always a full matrix algebra for SDPs arising from (commutative or non-commutative) sum of squares (SOS) problems. For SDPs arising from non-commutative SOS and commutators problems, the situation is less clear.We identify an exceptional case, where the corresponding matrix algebra is not the full matrix algebra, and use it to reprove the Burgdorf–Klep non-commutative variant of Hilbert's ternary quartics theorem: a bivariate non-commutative polynomial of degree at most 4 is trace positive if it is a sum of four squares and commutators.