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blocks have 1/2 canonical blocks canonical form canonical matrices center block characteristic polynomial Corollary corresponding factor define vj diag diagonal blocks direct sum DOCTOR OF PHILOSOPHY elementary divisors exists a standard exists a vector finite number form a standard hence imply index 4k invariant subspace iy(j J-isotropic subspace J-Lorentzian matrix J-orthogonal complement J-orthonormal basis J-skew matrix J-symmetric and J-skew Lemma 9 MacDuffee MATRICES TO CANONICAL minimum polynomial nilpotent J-skew matrices non-principal diagonal non-singular matrix non-zero J-value obtain a vector odd order orthogonal matrix permutation matrix positive definite principal diagonal proof of Theorem pure imaginary characteristic real vectors reduction of J-symmetric secondary diagonal set of index set of real standard set symmetric and skew T-symmetric matrix Theorem 13 thesis University of Wisconsin vector in N((A vectors of index vk+1 vk+2 yx Jy zeros elsewhere