Contractive determinantal representations of stable polynomials on a matrix polyball Academic Article

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abstract

• We show that a polynomial p with no zeros on the closure of a matrix unit polyball, a.k.a. a cartesian product of Cartan domains of type I, and such that $$p(0)=1$$, admits a strictly contractive determinantal representation, i.e., $$p=\det (I-KZ_n)$$, where $$n=(n_1,\ldots ,n_k)$$ is a k-tuple of nonnegative integers, $$Z_n=\bigoplus _{r=1}^k(Z^{(r)}\otimes I_{n_r})$$, $$Z^{(r)}=[z^{(r)}_{ij}]$$ are complex matrices, p is a polynomial in the matrix entries $$z^{(r)}_{ij}$$, and K is a strictly contractive matrix. This result is obtained via a noncommutative lifting and a theorem on the singularities of minimal noncommutative structured system realizations.

publication date

• January 1, 2016