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In control theory and in particular when studying the controllability 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
The Hautus lemma 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
- 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.