Abstract
This paper concerns a new generalized inverse for matrices of an arbitrary index. It is proved that every complex square matrix A possesses a {1,2,3}-inverse X such that XAm+1=Am for some integer m. We shall call such X a {1,2,3,1m}-inverse of A. A notable result is that A has a unique {1,2,3,1m}-inverse if and only if it has index 0 or 1, in which case ▪ is exactly its unique {1,2,3,1m}-inverse. For a matrix with an arbitrary index, the set of all its {1,2,3,1m}-inverses is completely determined. Some new characterizations of EP matrices, generalized EP matrices and m-EP matrices are established by using their {1,2,3,1m}-inverses.
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