Minimal Realizations and Determinantal Representations in the Indefinite Setting

Abstract

For a signature matrix J, we show that a rational matrix function M(z) that is strictly J-contractive on the unit circle ${{\mathbb {T}}}$, has a strict ${\tilde{J}}\oplus J$-contractive realization $\begin{bmatrix} A &{} B \\ C &{} D \end{bmatrix}$ for an appropriate signature matrix ${\tilde{J}}$; that is, $M(z) = D +zC (I -zA)^{-1} B$. As an application, we use this result to show that a two variable polynomial $p(z_1,z_2)$ of degree $(n_1,n_2)$, $n_2=1$, without roots on $\{ (0,0) \} \cup ({{\mathbb {T}}} \times \{ 0 \} ) \cup {{\mathbb {T}}}^2$ allows a determinantal representation

$$\begin{aligned} p(z_1, z_2) = p(0,0) \det (I_{n_1+1} - K Z), \ \ Z = z_1 I_{n_1} \oplus z_2 I_{n_2} , \end{aligned}$$

(1)

where K is a strict ${\tilde{J}}\oplus J$-contraction. This provides first evidence of a new conjecture that a two variable polynomial $p(z_1,z_2)$ of degree $(n_1,n_2)$ has a determinantal representation (1) with K a strict ${\tilde{J}}\oplus J$-contraction if and only if $p(z_1,z_2)$ has no roots in $\{ (0,0) \} \cup {{\mathbb {T}}}^2$.