  In this work we introduce and study novel Quasi Newton minimization methods
based on a Hessian approximation Broyden Class-\textit{type} updating scheme,
where a suitable matrix $\tilde{B}_k$ is updated instead of the current Hessian
approximation $B_k$. We identify conditions which imply the convergence of the
algorithm and, if exact line search is chosen, its quadratic termination. By a
remarkable connection between the projection operation and Krylov spaces, such
conditions can be ensured using low complexity matrices $\tilde{B}_k$ obtained
projecting $B_k$ onto algebras of matrices diagonalized by products of two or
three Householder matrices adaptively chosen step by step. Extended
experimental tests show that the introduction of the adaptive criterion, which
theoretically guarantees the convergence, considerably improves the robustness
of the minimization schemes when compared with a non-adaptive choice; moreover,
they show that the proposed methods could be particularly suitable to solve
large scale problems where $L$-$BFGS$ performs poorly.
