A note on the generalized Bott–Duffin inverse

By generalizing subspaces T,T⊥T,T⊥ found in [G. Chen, G. Liu, Y. Xue, Perturbation analysis of the generalized Bott–Duffin inverse of LL-zero matrices, Linear Multilinear Algebra 51 (2003) 11–20] to subspaces T,S⊂CnT,S⊂Cn, for which T⊕S=CnT⊕S=Cn, we get that if AT∩S=0AT∩S=0, AT,S(+)=PT,S(APT,S+PS,T)+ is (PT,SAPT,S)+(PT,SAPT,S)+, then S=T⊥S=T⊥, that is to say AA is a TT-zero matrix (i.e. AT∩T⊥=0AT∩T⊥=0).