Geodesic
Some equalities for generalized inverses of matrix sums and block circulant matrices
Archivum mathematicum, Tome 37 (2001) no. 4, pp. 301-306

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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.
Classification : 15A09, 15A23
Keywords: block circulant matrix; Moore-Penrose inverse; Drazin inverse; weighted Moore-Penrose inverse; quaternionic matrix 
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     author = {Tian, Yongge},
     title = {Some equalities for generalized inverses of matrix sums and block circulant matrices},
     journal = {Archivum mathematicum},
     pages = {301--306},
     publisher = {mathdoc},
     volume = {37},
     number = {4},
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     zbl = {1090.15005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2001__37_4_a6/}
}
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Tian, Yongge. Some equalities for generalized inverses of matrix sums and block circulant matrices. Archivum mathematicum, Tome 37 (2001) no. 4, pp. 301-306. http://geodesic.mathdoc.fr/item/ARM_2001__37_4_a6/