A counterexample to Kippenhahn's conjecture on Hermitian pencils (Q793811)

\textit{R. Kippenhahn} [Math. Nachr. 6, 193-228 (1951; Zbl 0044.162)] has conjectured that if the characteristic polynomial \(\det(xH+yK-zI)\) of the pencil \(xH+yK\) (where H,K are \(n\times n\) Hermitian matrices) contains a repeated factor, then H,K do not generate \(M_ n(C)\). He also verified the conjecture when the minimal polynomial of \(xH+yK\) has degree 1 or 2. Recently, \textit{H. Shapiro} [Linear Algebra Appl. 43, 201-221 (1982; Zbl 0487.15006)] proved the conjecture for \(n\leq 5.\) In this work, a pair of 8\(\times 8\) Hermitian matrices H,K is constructed such that H,K generate \(M_ 8(C)\) and the minimal polynomial of \(xH+yK\) has degree 4 showing that the conjecture is not true in general.