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

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*Yongge Tian*

**Address.** Department of Mathematics and Statistics, Queen's University,
Kingston, Ontario, CANADA K7L 3N6
**E-mail:** ytian@mast.queensu.ca

**Abstract.** 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.

**AMSclassification.** 15A09, 15A23

**Keywords.** Block circulant matrix, Moore-Penrose inverse, Drazin
inverse, weighted Moore-Penrose inverse,

quaternionic matrix