Ela on the Group Inverse of Linear Combinations of Two Group Invertible Matrices
نویسندگان
چکیده
hold. If such matrix X exists, then it is unique, denoted by A, and called the group inverse of A. It is well known that the group inverse of a square matrix A exists if and only if rank(A) = rank(A) (see, for example, [1, Section 4.4] for details). Clearly, not every matrix is group invertible. It is straightforward to prove that A is group invertible if and only if A is group invertible, and in this case, one has (A) = (A). Also, it should be evident that if A ∈ C and S ∈ C is nonsingular, then A is group invertible if and only if SAS is group invertible, and in this situation, one has (SAS) = SAS.
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