Negative answer to the conjecture on the convexity of the greatest eigenvalue (Q1851392)

The conjecture of Seeger [cf. \textit{D. Ye}, Linear Algebra Appl. 169, 103--109 (1992; Zbl 0756.15017)] on the convexity of the greatest eigenvalue \(\lambda(u)\) of a real symmetric matrix-valued function \(A(u)=(\alpha_{ij}(u))_{n \times n}\) of \(u \in \mathbb{R}^m\) is discussed. When \(n \geq 3\), it is shown by giving a \(3 \times 3\) counterexample which can be embedded in larger matrices that this conjecture is not true.