  In a recent paper it was shown how a matrix S-lemma can be applied to
construct controllers from noisy data. The current paper complements these
results by proving a matrix version of the classical Finsler's lemma. This
matrix Finsler's lemma provides a tractable condition under which all matrix
solutions to a quadratic equality also satisfy a quadratic inequality. We will
apply this result to bridge known data-driven control design techniques for
both exact and noisy data, thereby revealing a more general theory. The result
is also applied to data-driven control of Lur'e systems.
