EP properties of (b, c)-invertible matrices
Michael P. Drazin · Linear and Multilinear Algebra · 2020
For any semigroup S and any a,b,c,y∈S, the author in 2012 defined a as having (b,c)-inverse y if y∈bSy∩ySc, yab = b and cay = c (any such y being unique). This simultaneously generalizes several known generalized inverses, such as the Moore-Penrose inverse A† of any given complex n×n matrix A, for which one may take a = A, b=c=A∗, y=A†. It is well known that the EP property A†A=AA† is equivalent to each of four inclusions between the range or null spaces of A or A∗, and it is shown that, more generally, ya = ay and the four corresponding inclusions for (b,c)-inverses, while not equivalent, are at least connected by six implications: in particular, ya = ay still implies all the four inclusions.