Parametrized solutions X of the system AXA=AEA and A^kEAX= XAEA^k for a matrix A having index k

D. E. FERREYRA; M. LATTANZI; F. E. LEVIS; N. THOME

ELECTRONIC JOURNAL OF LINEAR ALGEBRA

INT LINEAR ALGEBRA SOC

Lugar: Haifa; Año: 2019 vol. 35 p. 503 - 510

1081-3810

Let A and E be n x n given complex matrices. This paper provides a necessary and sucient condition for the solvability to the matrix equation system given by AXA = AEA and A^kEAX = XAEA^k, for k being the index of A. In addition, we derive its general solution in terms of a G-Drazin inverse of A. As consequences, we derive new representations for the set of all G-Drazin inverses; we also obtain some interesting applications to show the importance of the obtained formulas.