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.