# Hautus lemma

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 [1] and.[2] Today it can be found in most textbooks on control theory.

## The main result

The Hautus lemma says that given a square matrix $\mathbf{A}\in M_n(\Re)$ and a $\mathbf{B}\in M_{n\times m}(\Re)$ the following are equivalent:

1. The pair $(\mathbf{A},\mathbf{B})$ is controllable
2. For all $\lambda\in\mathbb{C}$ it holds that $\operatorname{rank}[\lambda \mathbf{I}-\mathbf{A},\mathbf{B}]=n$
3. For all $\lambda\in\mathbb{C}$ that are eigenvalues of $\mathbf{A}$ it holds that $\operatorname{rank}[\lambda \mathbf{I}-\mathbf{A},\mathbf{B}]=n$

## References

• 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.
1. ^ Belevitch, V. (1968). Classical Control Theory. San Francisco: Holden–Day.
2. ^ Popov, V. M. (1973). Hyperstability of Control Systems. Berlin: Springer-Verlag. p. 320.