Klymchuk T. Canonical matrices and their applications.

Several aspects of the theory of matrix equations over fields and skew fields and the theory of systems of linear and semilinear mappings are considered. L. Huang (2001) defined the notion of consimilarity of quaternion matrices and constructed a canonical form under consimilarity. Several authors applied his canonical form to quaternion matrix equations. But the notion of consimilarity used by Huang and his followers is not convenient enough. It is proved in the dissertation that all definitions of consimilarity, which are based on different involutional automorphisms, are turned into one another by re-chousing of imaginary units in the skew field of quaternions, and therefore one can use any of them. A more convenient definition of the consimilarity of quaternion matrices is proposed, which has admitted to simplify the basic formulas and construct a common theory for consimilarity and for similarity of quaternion matrices. In particular, the canonical form of each quaternion matrix under new consimilarity is a complex matrix, in contrast to Huang's canonical form, which is a quaternion matrix. The structure of solutions of some quaternion matrix equations is studied using the new canonical form.