  The projection lemma (often also referred to as the elimination lemma) is one
of the most powerful and useful tools in the context of linear matrix
inequalities for system analysis and control. In its traditional formulation,
the projection lemma only applies to strict inequalities, however, in many
applications we naturally encounter non-strict inequalities. As such, we
present, in this note, a non-strict projection lemma that generalizes both its
original strict formulation as well as an earlier non-strict version. We
demonstrate several applications of our result in robust
linear-matrix-inequality-based marginal stability analysis and stabilization, a
matrix S-lemma, which is useful in (direct) data-driven control applications,
and matrix dilation.
