Hurwitz-stable matrix

A square matrix $A$ is called a Hurwitz matrix if every eigenvalue of $A$ has strictly negative real part, that is,

:

for each eigenvalue $\backslash lambda\_i$. $A$ is also called a stable matrix, because then the differential equation

:$\backslash dot\; x\; =\; A\; x$

is asymptotically stable, that is, $x(t)\backslash to\; 0$ as $t\backslash to\backslash infty.$

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

:$\backslash dot\; x(t)=A\; x(t)\; +\; B\; u(t)$

:$y(t)=C\; x(t)\; +\; D\; u(t)\backslash ,$

has a Hurwitz transfer function.

Any hyperbolic fixed point (or equilibrium point) of a continuous dynamical system is locally asymptotically stable if and only if the Jacobian of the dynamical system is Hurwitz stable at the fixed point.

The Hurwitz stability matrix is a crucial part of control theory. A system is stable if its control matrix is a Hurwitz matrix. The negative real components of the eigenvalues of the matrix represent negative feedback. Similarly, a system is inherently unstable if any of the eigenvalues have positive real components, representing positive feedback.

• M-matrix
• Perron–Frobenius theorem, which shows that any Hurwitz matrix must have at least one negative entry
• Z-matrix

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• {{planetmath reference|urlname=HurwitzMatrix|title=Hurwitz matrix}}

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Category:Matrices (mathematics)