Upper and lower bounds for ranks of matrix expressions using generalized inverses Original Research Article

The maximal and minimal ranks of the matrix expression A1−B1X1C1−B2X2C2 with respect to X1 and X2 are presented. As applications, the maximal and minimal ranks of A1−B1XC1 subject to a consistent matrix equation B2XC2=A2 are also determined. In addition, the maximal and minimal ranks of the Schur complement D−CA−B with respect to the generalized inverse A− of A and their various consequences are also considered.