On the conjugate product of complex polynomial matrices

In this paper, we propose a new operator of conjugate product for complex polynomial matrices. Elementary transformations are first investigated for the conjugate product. It is shown that an arbitrary complex polynomial matrix can be converted into the so-called Smith normal form by elementary transformations in the framework of conjugate product. Then the concepts of greatest common divisors and coprimeness are proposed and investigated, and some necessary and sufficient conditions for the coprimeness are established. Finally, it is revealed that two complex matrices AA and BB are consimilar if and only if (sI−A)(sI−A) and (sI−B)(sI−B) are conequivalent. Such a fact implies that the Jordan form of a complex matrix AA under consimilarity may be obtained by analyzing the Smith normal form of (sI−A)(sI−A).