Unitary similarity of the determinantal representation of unitary bordering matrices

Let F(x,y,z) be a hyperbolic ternary form of degree n. The Helton-Vinnikov theorem asserts that there exists an n×n complex symmetric matrix S such that FS(x,y,z)=F(x,y,z), where the determinantal hyperbolic ternary form of an n×n matrix B is defined by FB(x,y,z)=det⁡(xℜ(B)+yℑ(B)+zIn). Let A be an n×n unitary bordering matrix. It is known that A is unitarily similar to a symmetric matrix. In this paper, we investigate the unitary similarity between the unitary bordering matrix A and the Helton-Vinnikov symmetric matrix S admitted the determinantal representation of the ternary form FA(x,y,z) satisfying FS(x,y,z)=FA(x,y,z) for n=3,4.