Construction of determinantal representation of trigonometric polynomials
For a pair of \(n\times n\) Hermitian matrices \(S_{1}\) and \(S_{2}\), a real ternary homogeneous polynomial defined by \[ F(t, x,y)=\text{det}(tI_{n}+xS_{1}+yS_{2})\tag{1} \] is hyperbolic with respect to \((1,0,0)\). The authors construct real symmetric matrices \(S_{1}\) and \(S_{2}\) satisfying (1) for a hyperbolic form \(F_{\phi}\) corresponding to a general trigonometric polynomial \(\phi\). A criterion for the associated form \(F_{\phi}\) to be hyperbolic with respect to \((1,0,0)\) is given.
scientific article; zbMATH DE number 5924537
scientific article