Further results on the m-weak core inverse

Abstract

This work establishes multiple novel representations for the m-weak core inverse, accompanied by proofs of their validity. Furthermore, we derive perturbation bounds and analyze continuity properties for this generalized inverse. By utilizing the m-weak core inverse, we characterize the unique solution to a constrained minimization problem in the Frobenius norm framework: $\min\left\| M^{m+1}X - M^{2m}(M^m)^\dagger B \right\|_F^2$, subject to the range constraint \(\mathcal{R}(X) \subseteq \mathcal{R}(M^k)\), where \(m \in \mathbb{N}\), \(B \in \mathbb{C}^{n \times q}\), \(M \in \mathbb{C}^{n \times n}\) and \(\text{rank}(M) = k\).