On bordering of regular matrices
\textit{K. Nomakuchi's} characterization of generalized inverses (\(g\)-inverses) of matrices over the field of complex numbers using bordered matrices [ibid. 33, 1--8 (1980; Zbl 0442.15002)] in generalized to matrices over an arbitrary ring. Necessary and sufficient conditions are given for a commutative ring \(R\) to be a ring over which every regular matrix can be completed to an invertible matrix of a particular size by bordering. More precisely, for a regular \(A\), if \(G_{n\times m}\) is a \(g\)-inverse of \(A\), i.e., if \(AGA= A\), then \(\ker(A)= \text{Range}(I-GA)\) and \(\text{coker}(A)= \text{Range}(I-AG)\), and if \(I_n= AG= B_{m\times (m-r)} C_{(m-r)\times m}\) and \(I_m-GA= P_{n\times (n-r)} Q_{(n-r)\times n}\), then \((\begin{smallmatrix} A\\ Q\end{smallmatrix} \begin{smallmatrix} B\\ 0\end{smallmatrix})\) gives an invertible bordering of \(A\). In other words, a regular matrix \(A\) over \(R\) has nomempty set of invertible matrices \[ {\mathcal B}(A)= \{T= (\begin{smallmatrix} A\\ Q\end{smallmatrix} \begin{smallmatrix} P\\ R\end{smallmatrix})\} \] if and only if its kernel and cokernel are free. Such rings are precisely the projective free rings. Also, over such rings every regular matrix has a rank factorization. Using the bordering technique, the authors give an interesting method of computing minors of a reflexive \(g\)-inverse \(G\) of a regular matrix \(A\) when \(I-AG\) and \(I-GA\) have factorizations.
scientific article; zbMATH DE number 847504
scientific article