Outer inverse restricted by a linear system

We follow the idea to find solution of quadratic minimization problems restricted by linear constraints and to investigate underlying generalized inverses. A generalization of this approach leads to a new kind of outer generalized inverse, called the T-minimal G-constrained inverse of A (called the minimal (T,G) inverse of A shortly), with respect to a positive semidefinite/definite matrix T and a singular matrix G. The introduced generalized inverse is related with the minimization of the quadratic form 〈x, Tx〉 under the constraint set defined by the consistent equation GAx = Gb. In this way, the generalized inverse arising from this constrained quadratic minimization problem gives a minimal T-seminorm/norm solution of the system of linear equations GAx = Gb. The range and the null space of defined outer inverse are determined. Computational algorithms are defined and numerical experiments related with properties of the minimal (T,G) inverse are also presented