Hautus lemma

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

The main result[edit]

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


  • 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.