  The celebrated minimum residual method (MINRES), proposed in the seminal
paper of Paige and Saunders, has seen great success and wide-spread use in
solving linear least-squared problems involving Hermitian matrices, with
further extensions to complex symmetric settings. Unless the system is
consistent whereby the right-hand side vector lies in the range of the matrix,
MINRES is not guaranteed to obtain the pseudo-inverse solution. Variants of
MINRES, such as MINRES-QLP, which can achieve such minimum norm solutions, are
known to be both computationally expensive and challenging to implement. We
propose a novel and remarkably simple lifting strategy that seamlessly
integrates with the final MINRES iteration, enabling us to obtain the minimum
norm solution with negligible additional computational costs. We study our
lifting strategy in a diverse range of settings encompassing Hermitian and
complex symmetric systems as well as those with semi-definite preconditioners.
We also provide numerical experiments to support our analysis and showcase the
effects of our lifting strategy.
