A New Outer Inverse of a Matrix and Its Characterizations

This paper introduces a new outer inverse known as the Drazin-secondary generalized inverse (D-sg inverse), which combines the properties of the Drazin inverse and the secondary generalized inverse. We provide a representation of the D-sg inverse emphasizing its specific column space and row space characteristics. Several critical characterizations of the D-sg inverse are derived, demonstrating its significance. Additionally, we explore an application of the D-sg inverse in solving systems of linear equations, illustrating its practical utility in this context.