Weighted spectral core inverse

Based on the sum and difference of some generalized inverses, the spectral core inverse was introduced for square matrices and has some spectral properties of the Drazin inverse, which implies its significance and applications. In terms of a weight matrix, we extend the concept of the spectral core inverse to rectangular matrices and define the weighted spectral core inverse as a novel wider kind of weighted generalized inverses. We develop properties and characterizations of the weighted spectral core inverse. Representations and canonical forms for the weighted spectral core inverse are presented. We also consider the weighted dual spectral core inverse. Applying the weighted (dual) spectral core inverse, we solve minimization problems and systems of linear equations.