In this paper, we present a weak version (called weak EC inverse) of the recently defined EC inverse of a complex square matrix. More precisely, for any \(m\times n\) and \(n\times m\) complex matrices A and X, respectively, it is proved that there exists a unique complex matrix Y of size \(n\times m\) that satisfies \(AYA=A\) , \(YAY=Y\) , \((AY)^*=AY\) and \(YA=A^\dag A+XA-A^\dag AXA\) . When \(X=A^\dag \) , such a matrix Y reduces to the Moore-Penrose inverse. When A is a square complex matrix and X is a minimal rank weak Drazin inverse of A, different properties and representations of the weak EC inverse are developed. In particular, when X is the Drazin inverse, a new inverse (called spectral core inverse) with some spectral properties is presented as a special case of the weak EC inverse, which is expressed as sum and difference of the Moore-Penrose, DMP and CMP inverses. Also, applications of the weak EC inverse in solving minimizations problems and linear equations are obtained.
