On General Form of Restricted Generalized-Inverses
HE Chu-ning · Hu'nan Shifan Daxue xuebao. Ziran kexue ban · 2010
For A∈Cm×n,the four matrix equations (1)AGA=A,(2)GAG=G,(3)(AG)*=AG,(4)(GA)*=GAare called Penrose-equations.If G satisfies equations(i),(j),…,etc,then G is called an(ij…)-inverse of A and denoted by A(ij…),A{ij…} denotes the set of(ij…)-inverses.Given E∈Cp×n,F∈Cp×m,let S={X∈Cm×m|EX=F},A{i,j,…,E,F}=A{i,j,…}∩S.The necessary and sufficient conditions for 5 sets A{1,E,F},A{3,E,F},A{4,E,F},A{1,3,E,F} and A{1,4,E,F} to be nonempty are presented.Two sufficient conditions of the set A{1,2,E,F} to be nonempty are also given.Some general forms of these restricted generalized-inverses are put forward.