Constructions and characterizations of mixed reverse-order laws for the Moore–Penrose inverse and group inverse

This paper is concerned with constructions and characterizations of matrix equalities that involve mixed products of Moore–Penrose inverses and group inverses of two matrices. We first construct a mixed reverse-order law ( A ⁢ B ) † = B ∗ ⁢ ( A ∗ ⁢ A ⁢ B ⁢ B ∗ ) # ⁢ A ∗ {(AB)^{{\dagger}}=B^{\ast}(A^{\ast}ABB^{\ast})^{\#}A^{\ast}} , and show that this matrix equality always holds through the use of a special matrix rank equality and some matrix range operations, where A and B are two matrices of appropriate sizes, ( ⋅ ) ∗ {(\,\cdot\,)^{\ast}} , ( ⋅ ) † {(\,\cdot\,)^{{\dagger}}} and ( ⋅ ) # {(\,\cdot\,)^{\#}} mean the conjugate transpose, the Moore–Penrose inverse, and the group inverse of a matrix, respectively. We then give a diverse range of variation forms of this equality, and derive necessary and sufficient conditions for them to hold. Especially, we show an interesting fact that the two reverse-order laws ( A ⁢ B ) † = B † ⁢ A † {(AB)^{{\dagger}}=B^{{\dagger}}A^{{\dagger}}} and ( A ∗ ⁢ A ⁢ B ⁢ B ∗ ) # = ( B ⁢ B ∗ ) # ⁢ ( A ∗ ⁢ A ) # {(A^{\ast}ABB^{\ast})^{\#}=(BB^{\ast})^{\#}(A^{\ast}A)^{\#}} are equivalent.