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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 [http://www.win.tue.nl/~wscomalo/ Malo Hautus], can prove to be a powerful tool. This result appeared first in <ref>{{cite book|last=Belevitch|first=V.|title=Classical Control Theory|year=1968|publisher=Holden–Day|location=San Francisco}}</ref> and.<ref>{{cite book|last=Popov|first=V. M.|title=Hyperstability of Control Systems|year=1973|publisher=Springer-Verlag|location=Berlin|pages=320}}</ref> Today it can be found in most textbooks on control theory.

==The main result==
The Hautus lemma says that given a square matrix <math>\mathbf{A}\in M_n(\Re)</math> and a <math>\mathbf{B}\in M_{n\times m}(\Re)</math> the following are equivalent:
# The pair <math>(\mathbf{A},\mathbf{B})</math> is [[controllable]]
# For all <math>\lambda\in\mathbb{C}</math> it holds that <math>\operatorname{rank}[\lambda \mathbf{I}-\mathbf{A},\mathbf{B}]=n</math>
# For all <math>\lambda\in\mathbb{C}</math> that are eigenvalues of <math>\mathbf{A}</math> it holds that <math>\operatorname{rank}[\lambda \mathbf{I}-\mathbf{A},\mathbf{B}]=n</math>

==References==
*{{cite book|last=Sontag|first=Eduard D.|title=Mathematical Control Theory: Deterministic Finite-Dimensional Systems.|year=1998|publisher=Springer|location=New York|isbn=0-387-98489-5}}
*{{cite book|last=Zabczyk|first=Jerzy|title=Mathematical Control Theory – An introduction|year=1995|publisher=Birkhauser|location=Boston|isbn=3-7643-3645-5}}

<references/>

[[Category:Control theory]]
[[Category:Lemmas]]

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