On A Conjecture Of Kippenhahn About The Characteristic Polynomial Of A Pencil Generated By Two Hermitian Matrices. I

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Linear Algebra And Its Applications


Let A be an n × n complex matrix, and write A = H + iK, where i² = −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⁻¹AU 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.