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

Document Type


Publication Date


Published In

Linear Algebra And Its Applications


Let A be an n × n matrix; write A = H+iK, where i² = —1 and H and K are Hermitian. Let f(x,y,z) = det(zI−xH−yK). We first show that a pair of matrices over an algebraically closed field, which satisfy quadratic polynomials, can be put into block, upper triangular form, with diagonal blocks of size 1×1 or 2×2, via a simultaneous similarity. This is used to prove that if f(x,y,z) = [g(x,y,z)]ⁿ², where g has degree 2, then for some unitary matrix U, the matrix U∗AU is the direct sum of copies of a 2×2 matrix A₁, where A₁ is determined, up to unitary similarity, by the polynomial g(x,y,z). We use the connection between f(x,y,z) and the numerical range of A to investigate the case where f(x,y,z) has the form (z−αax− βy)ʳ[g(x,y,z)]⁵, where g(x,y,z) is irreducible of degree 2.