Maximal and Minimal Ranks of the Common Solution of Some Linear Matrix Equations over an Arbitrary Division Ring with Applications

摘要

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A_1X = C_1, XB_2 = C_2, A_3XB_3 = C_3 and A_4XB_4 = C_4 over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.