Let $F$ be a field, $V$ is a vector space on $F$, $\alpha$ is a linear transformation on $V$ and m is a positive integer greater than 1. If a linear transformation $\chi$ on $V$ satisfies ${\chi }^m=\alpha$, then $\chi$ is called an m-th root of $\alpha$. In this paper, using the Prüfer-Baer theorem of bounded modules over the principal ideal domain, we study the linear transformation which is algebraically on infinite dimensional complex vector spaces, and give the local and global conditions under which an m-th root of $\alpha$ can be expressed as a polynomial in $\alpha$. The obtained results deepen the related results in the matrix analysis.

In the report, a new proved method of the orthogonal canonical form theorem of real antisymmetric matrices was proposed, and some applications of the orthogonal canonical form of real antisymmetric matrices were further studied. Firstly, the centralizer and dimension of the real antisymmetric matrix were obtained; secondly, it was 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 were intuitively solved from the orthogonal canonical form.