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Some New Inequalities on Geometric Multiplicities and Gersgorin Discs

In this paper the authors continue their work on geometric multiplic- ities and Gersgorin discs done in a series of four recent papers. The new results involve principal submatrices, an upper bound on the absolute value of an eigenvalue, the rank of a matrix, non-real eigenvalues, and powers of matrices. Some consequences of the results and examples are provided.