Constructions and characterizations of mixed reverse-order laws for the Moore–Penrose inverse and group inverse
Yongge Tian · Georgian Mathematical Journal · 2024
Abstract 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 ∗ ) # = (