[For the entire collection see Zbl 0562.00008.]
Let C be a complex \(n\times n\) matrix such that \(x^*Cx\neq 0\) for \(x\in {\mathbb{C}}^ n\setminus \{0\}\). Then there exists a matrix Y such that \(Y^*CY\) is a diagonal unitary matrix. The author shows that \(\inf \{| x^*Cx|:x\in {\mathbb{R}}^ n,\| x\| =1\}\leq \cos (\theta /2)\cdot \| Y^{-1}\|^ 2\), where \(\theta\) is the angle at the vertex of the cone \(\{x^*Cx:x\in {\mathbb{C}}^ n\}\). His interest in the question relates to the numerical solution of the generalized eigenvalue problem \(Ax=\lambda Bx\) with A, B Hermitian. In an earlier paper he has proposed an algorithm to determine whether zero belongs to the numerical range of \(A+iB\). The algorithm is not guaranteed to succeed and in the present paper the author draws on inference as to when it should be stopped.