Page 74 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 65

is a Z-matrix and is positive stable since σ(M) ={1,2,3} and thus i + (M) = 3.
The matrix M can indeed be written as
P

⎛ ⎞ ⎛ ⎞
100 211
⎜ ⎟ ⎜ ⎟
M = 3 ⎝ 010 ⎠ − ⎝ 011 ⎠
001 000
with 3 >ρ(P) = 2 since σ(P) ={0,1,2}.
Remark 16. Note that any M-matrix is a P-matrix.

4.2.11 Positive Semistable Matrix

We say that M is positive semistable if i − (M) = 0, that is,

(∀λ ∈ σ(M)) : re(λ) ≥ 0.
For example, the matrix

⎛ ⎞
2 −10
M = ⎝ 1 2 0 ⎠
⎜
⎟
0 0 3
is positive semistable since σ(M) ={2 − i,2 + i,3}.
Remark 17. Suppose that there exists a symmetric positive definite matrix G ∈
T
R n×n such that H = GM + M G is positive semidefinite. Then M is positive
n
n
semistable. Indeed, let λ ∈ σ(M) and x = a + bi ∈ C ,x
= 0 (a,b ∈ R ) be
T
such that Mx = λx.Weset S = GM − M G and note that H + S = 2GM.For
n
z ∈ C, we denote by z the complex conjugate of z, and for u,v ∈ C ,weset
u,v C = u 1 v 1 + u 2 v 2 + ... + u n v n .
n
We see that
(H + S)x = 2GMx = 2λGx,

and thus
(H + S)x,x C = 2λ Gx,x C .
n
n
The matrix S is skew-symmetric, and thus
T
n
Sx,x C = Sa,a + Sb,b + i( Sa,b −ÐSb,a ) = i Sa − S a,b
= 2i Sa,b .