Estimate for the lower bound of rank and the upper bound of eigenvalues normsʹ sum of given quaternion matrix

This work is concerned with estimating the lower bound of rank for a given quaternion square matrix, rectangular matrix or normal matrix, and estimating the upper bound for the sum of the eigenvalue norms of a given quaternion matrix. A sufficient condition is provided to confirm whether a given quaternion matrix is nonsingular. It sharpens some results due to Tu Bo-xun and Semyon Aranovich Ger²chgorin and Schur, respectively. Some examples are provided to show the effectiveness of our results.