A conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. II |
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Authors: | Helene Shapiro |
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Institution: | Department of Mathematics Swarthmore College Swarthmore, Pennsylvania 19081, USA |
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Abstract: | 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 , where g has degree 2, then for some unitary matrix U, the matrix U1AU 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)rg(x,y,z)]s, where g(x,y,z) is irreducible of degree 2. |
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