On computation of skew-symmetric generator for an orthogonal matrix

Iwo Biborski


In this paper, we constructively  prove that  for any matrix  A over a field of characteristic 0 and its eigenvalue λ ≠ 0 there exists a diagonal matrix D with diagonal coefficients ±1 such that DA has no eigenvalue λ. Hence and by the canonical result on Cayley transformation, for each orthogonal matrix U one can find a diagonal matrix D and a skew-symmetric matrix  S such that U = D(S − I )−1 (S + I ).

Słowa kluczowe: Orthogonal matrix, Cayley transformation, characteristic polynomial

1. Liebeck H., Osborne A., The generation of all rational orthogonal matrices, Amer. Math. Monthly, 98, No. 2 (Feb. 1991), 131-133.

2. Mirsky L., An introduction to linear algebra, Dover Publications, 1990.

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