Abstract Let A be an n × n complex matrix, and write A = H + iK, where i 2 = −1 and H and K are Hermitian matrices. The characteristic polynomial of the pencil xH + yK is f( x, y, z) = det( zI − xH − yK). Suppose f( x, y, z) is factored into a product of irreducible polynomials. Kippenhahn [5, p. 212] conjectured that if there is a repeated factor, then there is a unitary matrix U such that U −1 AU is block diagonal. We prove that if f( x, y, z) has a linear factor of multiplicity greater than n⧸3, then H and K have a common eigenvector. This may be viewed as a special case of Kippenhahn’s conjecture.