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In control theory and in particular when studying the properties of a linear time-invariant system in state space form, the Hautus lemma, named after Malo Hautus, 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
Hautus Lemma for detectability
- Sontag, Eduard D. (1998). Mathematical Control Theory: Deterministic Finite-Dimensional Systems. New York: Springer. ISBN 0-387-98489-5.
- Zabczyk, Jerzy (1995). Mathematical Control Theory – An introduction. Boston: Birkhauser. ISBN 3-7643-3645-5.
- Belevitch, V. (1968). Classical Control Theory. San Francisco: Holden–Day.
- Popov, V. M. (1973). Hyperstability of Control Systems. Berlin: Springer-Verlag. p. 320.