In mathematics, the Routh–Hurwitz matrix, or more commonly just Hurwitz matrix, corresponding to a polynomial is a particular matrix whose nonzero entries are coefficients of the polynomial.
Contents
Hurwitz matrix and the Hurwitz stability criterion
Namely, given a real polynomial
p
(
z
)
=
a
0
z
n
+
a
1
z
n
−
1
+
⋯
+
a
n
−
1
z
+
a
n
{\displaystyle p(z)=a_{0}z^{n}+a_{1}z^{n-1}+\cdots +a_{n-1}z+a_{n}}
the
n
×
n
{\displaystyle n\times n}
square matrix
H
=
(
a
1
a
3
a
5
…
…
…
0
0
0
a
0
a
2
a
4
⋮
⋮
⋮
0
a
1
a
3
⋮
⋮
⋮
⋮
a
0
a
2
⋱
0
⋮
⋮
⋮
0
a
1
⋱
a
n
⋮
⋮
⋮
⋮
a
0
⋱
a
n
−
1
0
⋮
⋮
⋮
0
a
n
−
2
a
n
⋮
⋮
⋮
⋮
a
n
−
3
a
n
−
1
0
0
0
0
…
…
…
a
n
−
4
a
n
−
2
a
n
)
.
{\displaystyle H={\begin{pmatrix}a_{1}&a_{3}&a_{5}&\dots &\dots &\dots &0&0&0\\a_{0}&a_{2}&a_{4}&&&&\vdots &\vdots &\vdots \\0&a_{1}&a_{3}&&&&\vdots &\vdots &\vdots \\\vdots &a_{0}&a_{2}&\ddots &&&0&\vdots &\vdots \\\vdots &0&a_{1}&&\ddots &&a_{n}&\vdots &\vdots \\\vdots &\vdots &a_{0}&&&\ddots &a_{n-1}&0&\vdots \\\vdots &\vdots &0&&&&a_{n-2}&a_{n}&\vdots \\\vdots &\vdots &\vdots &&&&a_{n-3}&a_{n-1}&0\\0&0&0&\dots &\dots &\dots &a_{n-4}&a_{n-2}&a_{n}\end{pmatrix}}.}
is called Hurwitz matrix corresponding to the polynomial
p
{\displaystyle p}
. It was established by Adolf Hurwitz in 1895 that a real polynomial with
a
0
>
0
{\displaystyle a_{0}>0}
is stable
(that is, all its roots have strictly negative real part) if and only if all the leading principal minors of the matrix
H
(
p
)
{\displaystyle H(p)}
are positive:
Δ
1
(
p
)
=
|
a
1
|
=
a
1
>
0
Δ
2
(
p
)
=
|
a
1
a
3
a
0
a
2
|
=
a
2
a
1
−
a
0
a
3
>
0
Δ
3
(
p
)
=
|
a
1
a
3
a
5
a
0
a
2
a
4
0
a
1
a
3
|
=
a
3
Δ
2
−
a
1
(
a
1
a
4
−
a
0
a
5
)
>
0
{\displaystyle {\begin{aligned}\Delta _{1}(p)&={\begin{vmatrix}a_{1}\end{vmatrix}}&&=a_{1}>0\\[2mm]\Delta _{2}(p)&={\begin{vmatrix}a_{1}&a_{3}\\a_{0}&a_{2}\\\end{vmatrix}}&&=a_{2}a_{1}-a_{0}a_{3}>0\\[2mm]\Delta _{3}(p)&={\begin{vmatrix}a_{1}&a_{3}&a_{5}\\a_{0}&a_{2}&a_{4}\\0&a_{1}&a_{3}\\\end{vmatrix}}&&=a_{3}\Delta _{2}-a_{1}(a_{1}a_{4}-a_{0}a_{5})>0\end{aligned}}}
and so on. The minors
Δ
k
(
p
)
{\displaystyle \Delta _{k}(p)}
are called the Hurwitz determinants. Similarly, if
a
0
<
0
{\displaystyle a_{0}<0}
then the polynomial is stable if and only if the principal minors have alternating signs starting with a negative one.
Example
As an example, consider the matrix
M
=
(
−
1
−
1
0
1
−
1
0
0
0
−
1
)
,
{\displaystyle M={\begin{pmatrix}-1&-1&0\\1&-1&0\\0&0&-1\end{pmatrix}},}
and let
p
(
x
)
=
det
(
x
I
−
M
)
=
|
x