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The maximal and minimal ranks of \(A - BXC\) with applications. (English) Zbl 1036.15004
Maximal and minimal ranks of a linear matrix expression \(A - BXC\) over an arbitrary field \(\mathbb{F}\), where \(A \in {\mathbb F}^{m \times n}\), \(B \in {\mathbb F}^{m \times k}\) and \(C \in {\mathbb F}^{l \times n}\), are investigated with regard to a variant matrix \(X \in {\mathbb F}^{k \times l}\). The obtained results on the extremal ranks of the given linear matrix expression can be used for finding extremal ranks of other linear and nonlinear matrix expressions. The rank invariance and the range invariance of the expression \(A - BXC\) with respect to its variant matrix \(X\) is discussed. On the basis of generalized inverses of matrices the general solution to the rank equation \(\text{rank}(A- BXC)+ \text{rank}(BXC)= \text{rank}(A)\) is presented and the minimal rank of \(A - BXC\) subject to this equation is obtained.

MSC:
15A03 Vector spaces, linear dependence, rank, lineability
15A09 Theory of matrix inversion and generalized inverses
15A24 Matrix equations and identities
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