Rank equalities related to the generalized inverses \(A_{T,S}^{(2)}\), \(B_{T_1,S_1}^{(2)}\) of two matrices \(A\) and \(B\)
Let \(A\) be an \(m\times n\) matrix over \(\mathbb{C}\). A \{2\}-inverse of \(A\) is an \(n\times m\) matrix \(X\) such that \(XAX=X\) [see \textit{A. Ben-Israel} and \textit{T. N. E. Greville}, Generalized inverses: Theory and applications (Wiley, New York) (1974; Zbl 0305.15001) for background on generalized matrix inverses]. Let \(T\) and \(S\) be subspaces of \(\mathbb{C}^{n}\) and \(\mathbb{C}^{m}\), respectively, with \(m-\dim S=\dim T\leq rank(A)\). Then there exists a unique \{2\}-inverse \(X\) of \(A\) with range \(T\) and null space \(S\) if and only if \(AT\oplus S=\mathbb{C}^{m}\); this inverse is denoted \(A_{T,S}^{(2)}\). More explicitly, if \(G\) is any \(n\times m\) matrix with range \(T\) and null space \(S\), and \((AG)_{g}\) denotes the group inverse of \(AG\), then \(A_{T,S}^{(2)}=G(AG)_{g}\). For specific choices of \(S\) and \(T\), \(A_{T,S}^{(2)}\) is equal to the Moore-Penrose inverse, the group inverse, the Drazin inverse, and various other generalized inverses. The authors prove some inequalities relating to the ranks of matrices and their generalized inverses. The following is typical. Let \(A\) and \(B\) be matrices of dimensions \(m\times n\) and \(m\times k\), respectively, and \(G_{a}\) and \(G_{b}\) be of dimensions \(n\times m\) and \(k\times m\), respectively. Suppose that the range and null space of \(G_{a}\) (respectively \(G_{b}\)) are \(T\) and \(S\) (\(T_{1}\) and \(S_{1}\)) and that the generalized inverses \(A_{T,S}^{(2)}\) and \(B_{T_{1},S_{1}}^{(2)}\) exist. Then \(rank(AA_{T,S} ^{(2)}-BB_{T_{1},S_{1}}^{(2)})=rank(G_{a}^{T},G_{b}^{T})+rank(AG_{a} ,BG_{b})-rank(G_{a})-rank(G_{b})\).
scientific article; zbMATH DE number 2128787