In mathematics, a Hurwitz-stable matrix, or more commonly simply Hurwitz matrix, is a square matrix whose eigenvalues all have strictly negative real part. Some authors also use the term stability matrix. Such matrices play an important role in control theory.
Definition A square matrix A {\displaystyle A} is called a Hurwitz matrix if every eigenvalue of A {\displaystyle A} has strictly negative real part, that is,
Re [ λ i ] < 0 {\displaystyle \operatorname {Re} [\lambda _{i}]<0\,}
for each eigenvalue λ i {\displaystyle \lambda _{i}} . A {\displaystyle A} is also called a stable matrix, because then the differential equation
x ˙ = A x {\displaystyle {\dot {x}}=Ax}
is asymptotically stable, that is, x ( t ) → 0 {\displaystyle x(t)\to 0} as t → ∞ . {\displaystyle t\to \infty .}
If G ( s ) {\displaystyle G(s)} is a (matrix-valued) transfer function, then G {\displaystyle G} is called Hurwitz if the poles of all elements of G {\displaystyle G} have negative real part. Note that it is not necessary that G ( s ) , {\displaystyle G(s),} for a specific argument s , {\displaystyle s,} be a Hurwitz matrix — it need not even be square. The connection is that if A {\displaystyle A} is a Hurwitz matrix, then the dynamical system
x ˙ ( t ) = A x ( t ) + B u ( t ) {\displaystyle {\dot {x}}(t)=Ax(t)+Bu(t)}
y ( t ) = C x ( t ) + D u ( t ) {\displaystyle y(t)=Cx(t)+Du(t)\,}
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.
See also M-matrix Perron–Frobenius theorem, which shows that any Hurwitz matrix must have at least one negative entry Z-matrix
References
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