Hurwitz matrix

In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial.
Hurwitz matrix and the Hurwitz stability criterion.
Namely, given a real polynomial
the formula_2 square matrix
is called Hurwitz matrix corresponding to the polynomial formula_4. It was established by Adolf Hurwitz in 1895 that a real polynomial with formula_5 is stable
(that is, all its roots have strictly negative real part) if and only if all the leading principal minors of the matrix formula_6 are positive:
and so on. The minors formula_8 are called the Hurwitz determinants. Similarly, if formula_9 then the polynomial is stable if and only if the principal minors have alternating signs starting with a negative one.
Hurwitz stable matrices.
In engineering and stability theory, a square matrix formula_10 is called a stable matrix (or sometimes a Hurwitz matrix) if every eigenvalue of formula_10 has strictly negative real part, that is,
for each eigenvalue formula_13. formula_10 is also called a stability matrix, because then the differential equation
is asymptotically stable, that is, formula_16 as formula_17
If formula_18 is a (matrix-valued) transfer function, then formula_19 is called Hurwitz if the poles of all elements of formula_19 have negative real part. Note that it is not necessary that formula_21 for a specific argument formula_22 be a Hurwitz matrix — it need not even be square. The connection is that if formula_10 is a Hurwitz matrix, then the dynamical system
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.