Upper and lower bounds for ranks of matrix expressions using generalized inverses (Q1855375)

The author studies the minimal and maximal possible ranks of the matrix expression \(A_1-B_1X_1C_1-B_2X_2C_2\) (with known matrices \(A_1\), \(B_1\), \(C_1\), \(B_2\), \(C_2\) and w.r.t. the variant matrices \(X_1\), \(X_2\)). As applications, the minimal and maximal ranks of \(A_1-B_1XC_1\) are determined when \(X\) satisfies the consistent matrix equation \(B_2XC_2=A_2\). A generalized inverse \(A^-\) of the matrix \(A\) is by definition any of the solutions to the equation \(AXA=A\). The author considers also the minimal and maximal possible ranks of the Schur complement \(D-CA^-B\) w.r.t. the generalized inverse \(A^-\) of \(A\).