On products of EP matrices
In this note the authors give necessary and sufficient conditions for a product of two \(n\times n\) complex EP matrices to be EP. All matrices will be complex, and rank, ranges, and row spaces are denoted by \(r(\cdot)\), \(R(\cdot)\), and \(RS(\cdot)\), respectively. A matrix is called EP if \(R(A)= R(A^*)\) and that the group inverse \(A^\sharp\) of \(A\) exists exactly when \(r(A)= r(A^2)\). The main theorem is as follows: THEOREM. Let \(A,B\in C_{n\times n}\) be EP. Then the following are equivalent: i) \(R(AB)= R(A)\cap R(B)\) and \(RS(AB)= RS(A)\cap RS(B)\). ii) \(R(AB)\subset R(B)\) and \(RS(AB)\subset R(A).\) iii) \(AB\) is EP.
scientific article; zbMATH DE number 991128
scientific article