三个矩阵乘积厄米特Moore-Penrose逆的等式与不等式

We investigate relationships between the Moore-Penrose inverse(ABA*)and the product (AB)(1,2,3)]*B(AB)(1,2,3)through some rank and inertia formulas for the difference of(ABA*)-(AB)(1,2,3)]*B(AB)(1,2,3),where B is Hermitian matrix and(AB)(1,2,3)is a {1,2,3}-inverse of AB.We show that there always exists an(AB)(1,2,3)such that(ABA*)= (AB)(1,2,3)]*B(AB)(1,2,3)holds.In addition,we also establish necessary and sufficient conditions for the two inequalities(ABA*) (AB)(1,2,3)]*B(AB)(1,2,3)and(ABA*)(AB)(1,2,3)]*B(AB)(1,2,3)to hold in the L¨owner partial ordering.Some variations of the equalities and inequalities are also presented.In particular,some equalities and inequalities for the Moore-Penrose inverse of the sum A + B of two Hermitian matrices A and B are established.

Some Equalities and Inequalities for the Hermitian Moore-Penrose Inverse of Triple Matrix Product with Applications

We investigate relationships between the Moore-Penrose inverse $(ABA^{*})^{\dag}$ and the product $(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ through some rank and inertia formulas for the difference of $(ABA^{*})^{\dag} - (AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$, where $B$ is Hermitian matrix and $(AB)^{(1,2,3)}$ is a $\{1,\, 2,\,3\}$-inverse of $AB$. We show that there always exists an $(AB)^{(1,2,3)}$ such that $(ABA^{*})^{\dag}=(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ holds. In addition, we also establish necessary and sufficient conditions for the two inequalities $(ABA^{*})^{\dag} \succ (AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ and $(ABA^{*})^{\dag} \prec (AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ to hold in the L\"owner partial ordering. Some variations of the equalities and inequalities are also presented. In particular, some equalities and inequalities for the Moore-Penrose inverse of the sum $A + B$ of two Hermitian matrices $A$ and $B$ are established.