Ranks of submatrices and the off-diagonal indices of a square matrix Original Research Article

In the first part of the paper we determine bounds for the ranks of certain submatrices of square matrices taken from a prescribed similarity class. Then we discuss the concept of off-diagonal indices (defined in the Introduction) which, very roughly speaking, measure, for each given integer s, how far we have to go off the main diagonal of a square matrix, to find an s×s nonzero minor. Some open problems are stated.