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- W2051925191 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 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 n2 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)]s, where g(x,y,z) is irreducible of degree 2." @default.
- W2051925191 created "2016-06-24" @default.
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- W2051925191 date "1982-06-01" @default.
- W2051925191 modified "2023-09-27" @default.
- W2051925191 title "A conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. II" @default.
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- W2051925191 doi "https://doi.org/10.1016/0024-3795(82)90212-9" @default.
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