Parameterization of real orthogonal antisymmetric matrices

In response to a question posed by João P. Silva, I demonstrate that an arbitrary 2n×2n real orthogonal antisymmetric matrix can be parameterized by n(n−1) continuous angular parameters. An algorithm is provided for constructing the general form for such a matrix. This algorithm is based on observation that Sp(n,R) ∩ O(2n) ∼= U(n) and employs the parameterization of the coset space SO(2n)/U(n). An explicit parameterization for a 4× 4 real orthogonal antisymmetric matrix is exhibited. 1 Decomposition of a real orthogonal antisymmetric matrix In these notes, I shall discuss the parameterization of an arbitrary real orthogonal antisymmetric matrix M , which satisfies M = −M , MM = I , (1) where I is the identity matrix. First, we note that M is a 2n× 2n nonsingular matrix such that detM = 1, where n can be any positive integer. Since MM = I, it follows that detM = ±1, which implies that that M is nonsingular. Hence, M is an even-dimensional matrix, since any odd-dimensional antisymmetric matrix M satisfies detM = 0. Moreover, for any even-dimensional 2n × 2n antisymmetric matrix M , the pfaffian of M , denoted by pfM , is defined by pfM = 1 2nn! ǫi1j1i2j2···injnMi1j1Mi2j2 · · ·Minjn , (2) where ǫ is the rank-2n Levi-Civita tensor, and the sum over repeated indices is implied. A well-known result states that for any antisymmetric matrix M , detM = [pf M ]. (3) In particular, if M is also orthogonal then detM = 1, in which case pf M = ±1. Next, we note that the eigenvalues of any real antisymmetric matrix M are purely imaginary. Moreover if λ is an eigenvalue of M then λ is also an eigenvalue (see, e.g., Ref. [2]). Thus, the Let M be a d× d antisymmetric matrix. Since det M = det (−MT) = det (−M) = (−1)d det M , it follows that det M = 0 if d is odd. For a discussion of the properties of the pfaffian, see, e.g., Ref. [1].