Reduction of Matrices to Canonical Form Under Generalized Lorentzian Transformations |
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a+ib A₁ Ay 2k canonical blocks canonical form center block characteristic polynomial characteristic vector corresponding Define V₁ diag diagonal blocks direct sum DOCTOR OF PHILOSOPHY elementary divisors exists a standard exists a vector form a standard hence imaginary characteristic root imply index 4k iy(j J-isotropic subspace J-Lorentzian matrix J-orthogonal complement J-orthonormal basis J-skew matrix J-symmetric and J-skew J-symmetric matrix Lemma MATRICES TO CANONICAL minimum polynomial nilpotent J-skew matrices non-singular matrix non-zero J-value obtain a vector odd order orthogonal matrix P₁ permutation matrix principal diagonal proof of Theorem pure imaginary characteristic real characteristic root real vectors reduction of J-symmetric secondary diagonal set of index set of real standard set symmetric and skew Theorem 13 v₁₂ V₂ vector in N((A vector u(0 vectors of index Y₂ zeros elsewhere μ μ μ χ μν