Some applications of canonical forms of real antisymmetric matrices

By giving a new proof of the orthogonal canonical form theorem of real antisymmetric matrices, some applications of the orthogonal canonical form of real antisymmetric matrices are further studied. Firstly, for any real antisymmetric matrix, its centralizer and its dimension are obtained. Secondly, it is proved that for each positive odd number m, the real antisymmetric matrix has only one m-th root of real antisymmetric matrix. Finally, some basic results about real antisymmetric matrices are intuitively solved from the orthogonal canonical form.