Numerical Range and a Generalization of Duffin’s Overdamping Criterion

The joint numerical range of tuples of matrices is a powerful tool for proving results which are useful in optimization, such as the \(\mathcal{S}\) -lemma. Here we provide a similar proof for another result, namely the equivalence of a certain positivity criterion to Duffin’s overdamping condition involving quadratic matrix-valued polynomials. We show how the proof is generalizable to higher degrees of matrix-valued polynomials.