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### Some equalities for generalized inverses of matrix sums and block circulant matrices

Archivum Mathematicum

Let ${A}_{1},{A}_{2},\cdots ,{A}_{n}$ be complex matrices of the same size. We show in this note that the Moore-Penrose inverse, the Drazin inverse and the weighted Moore-Penrose inverse of the sum ${\sum }_{t=1}^{n}{A}_{t}$ can all be determined by the block circulant matrix generated by ${A}_{1},{A}_{2},\cdots ,{A}_{n}$. In addition, some equalities are also presented for the Moore-Penrose inverse and the Drazin inverse of a quaternionic matrix.

### How to characterize commutativity equalities for Drazin inverses of matrices

Archivum Mathematicum

Necessary and sufficient conditions are presented for the commutativity equalities ${A}^{*}{A}^{D}={A}^{D}{A}^{*}$, ${A}^{†}{A}^{D}={A}^{D}{A}^{†}$, ${A}^{†}A{A}^{D}={A}^{D}A{A}^{†}$, $A{A}^{D}{A}^{*}={A}^{*}{A}^{D}A$ and so on to hold by using rank equalities of matrices. Some related topics are also examined.

### A new rank formula for idempotent matrices with applications

Commentationes Mathematicae Universitatis Carolinae

It is shown that $\text{rank}\left({P}^{*}AQ\right)=\text{rank}\left({P}^{*}A\right)+\text{rank}\left(AQ\right)-\text{rank}\left(A\right),$ where $A$ is idempotent, $\left[P,Q\right]$ has full row rank and ${P}^{*}Q=0$. Some applications of the rank formula to generalized inverses of matrices are also presented.

### Common solutions of a pair of matrix equations.

Applied Mathematics E-Notes [electronic only]

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