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Hurwitz Matrix

A square matrix is called a Hurwitz matrix if all eigenvalues of have strictly negative real part, $Re[\lambda_i] < 0$ ; is also called a stability matrix, because the feedback system

$\displaystyle \dot x = A x$

is stable.

If is a (matrix-valued) transfer function, then is called Hurwitz if the poles of all elements of have negative real part. Note that it is not necessary that , for a specific argument , be a Hurwitz matrix -- it need not even be square. The connection is that if is a Hurwitz matrix, then the dynamical system

$\displaystyle \dot x(t)$	$\displaystyle =$	$\displaystyle A x(t) + B u(t)$
$\displaystyle y(t)$	$\displaystyle =$	$\displaystyle C x(t) + D u(t)$

has a Hurwitz transfer function.

Reference: Hassan K. Khalil, Nonlinear Systems, Prentice Hall, 2002

1	Complex
1	Positive
1	Strict
1	Even Number
1	Argument
1	Necessary And Sufficient
2	Characteristic Values And Vectors Of A Matrix
2	Square Matrix
2	Bibliography For Topology
3	Special Elements In A Relation Algebra
4	Pole
6	Transfer Function
8	Dynamical System