Page 67 - Complementarity and Variational Inequalities in Electronics

58 Complementarity and Variational Inequalities in Electronics

where a 0 ,a 1 ,...,a n ∈ R, a 0 > 0. We denote by
(M) the Routh–Hurwitz ma-
trix associated with p, which is defined by

if 0 ≤ 2i − j ≤ n,
a 2i−j

(p) ij = (4.5)
0 if otherwise.
For example, if n = 5, then

⎛ ⎞
a 1 a 0 0 0 0
⎜ ⎟
⎜ a 3 a 2 a 1 a 0 0 ⎟
⎜ ⎟

(p) = ⎜ a 5 a 4 a 3 a 2 a 1 ⎟.
⎝ 0 0 a 5 a 4 a 3 ⎟
⎜
⎠
0 0 0 0 a 5
The following result gives a criterion that can be used to determine whether all
the roots of the polynomial p have negative real parts (see e.g. [54]).

Theorem 1 (Routh–Hurwitz criterion). All the roots of the equation

p(λ) = 0

have real negative parts if and only if the leading principal minors of the Routh–
Hurwitz matrix
(p) are positive.

Corollary 1. We have

i − (M) = n ⇔ 1 (
(p M )) > 0, 12 (
(p M )) > 0,..., 12...n (
(p M )) > 0,

where

(∀λ ∈ C) : p M (λ) = det(λI − M).

It is clear that i + (M) = n if and only if i − (−M) = n, and we get the follow-
ing result.

Corollary 2. We have

i + (M) = n ⇔ 1 (
(q M )) > 0, 12 (
(q M )) > 0,..., 12...n (
(q M )) > 0

where
(∀λ ∈ C) : q M (λ) = det(λI + M).