Page 72 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 63

Theorem 2. Let M ∈ R n×n . Then M is positive stable if and only if there exists
T
a symmetric and positive definite matrix G ∈ R n×n such that GM + M G is
positive definite.

If M is positive stable, then there exists a symmetric positive definite matrix
G ∈ R n×n such that

1 1
n
(∀x ∈ R ,x
= 0) : Mx,Gx = Mx,Gx + Gx,Mx
2 2
1 1 T 1 T
= GMx,x + M Gx,x = (GM + M G),x > 0.
2 2 2
The Routh–Hurwitz criterion can be used to check if the matrix M is positive
stable. We have indeed

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

with
(∀λ ∈ C) : q M (λ) = det(λI + M)

and where
(q M ) is the Routh–Hurwitz matrix associated with the polyno-
mial q M (see (4.5)). Note that q M = p −M is the characteristic polynomial
associated with the matrix −M.
For example, let us again consider the matrix M in (4.6).Wehave

2
3
p −M (λ) = λ + 3λ + 4λ + 2.
The corresponding Routh–Hurwitz matrix is given by
⎛ ⎞
3 1 0

(p −M ) = ⎝ 2 4 3 ⎠ .
⎟
⎜
0 0 2
We have 1 (
(p −M )) = 3, 12 (
(p −M )) = 9, 123 (
(p −M )) = 18, and the
matrix is thus positive stable. We have indeed σ(M) ={1 − i,1 + i,1}.

4.2.9 Z-Matrix

We say that M is a Z-matrix if

(∀i,j ∈{1,...,n},i
= j) : a ij ≤ 0.