# On a conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. I

- Authors
- Journal
- Linear Algebra and its Applications 0024-3795
- Publisher
- Elsevier
- Publication Date
- Volume
- 43
- Identifiers
- DOI: 10.1016/0024-3795(82)90254-3

## Abstract

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.

## There are no comments yet on this publication. Be the first to share your thoughts.