In control theory and in particular when studying the properties of a linear time-invariant system in state space form, the Hautus lemma (after Malo L. J. Hautus), also commonly known as the Popov-Belevitch-Hautus test or PBH test, can prove to be a powerful tool. This result appeared first in and. Today it can be found in most textbooks on control theory. The main result There exist multiple forms of the lemma. Hautus Lemma for controllability The Hautus lemma for controllability says that given a square matrix and a the following are equivalent: The pair is controllable For all it holds that For all that are eigenvalues of it holds that Hautus Lemma for stabilizability The Hautus lemma for stabilizability says that given a square matrix and a the following are equivalent: The pair is stabilizable For all that are eigenvalues of and for which it holds that Hautus Lemma for observability The Hautus lemma for observability says that given a square matrix and a the following are equivalent: The pair is observable For all it holds that For all that are eigenvalues of it holds that Hautus Lemma for detectability The Hautus lemma for detectability says that given a square matrix and a the following are equivalent: The pair is detectable For all that are eigenvalues of and for which it holds that References Control theory Lemmas