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Is there a small skew Cayley transform with zero diagonal?

Linear Algebra and its Applications
Publication Date
DOI: 10.1016/j.laa.2005.08.027
  • Cayley Transform
  • Skew Matrix
  • Zero Diagonal


Abstract The eigenvectors of an Hermitian matrix H are the columns of some complex unitary matrix Q. For any diagonal unitary matrix Ω the columns of Q · Ω are eigenvectors too. Among all such Q · Ω at least one has a skew-Hermitian Cayley transform S ≔ ( I + Q · Ω) −1 · ( I − Q · Ω) with just zeros on its diagonal. Why? The proof is unobvious, as is the further observation that Ω may also be so chosen that no element of this S need exceed 1 in magnitude. Thus, plausible constraints, easy to satisfy by perturbations of complex eigenvectors when Hermitian matrix H is perturbed infinitesimally, can be satisfied for discrete perturbations too. But if H is real symmetric, Q real orthogonal and Ω restricted to diagonals of ±1’s, then whether at least one real skew-symmetric S must have no element bigger than 1 in magnitude is not known yet.

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