Page 79 - Complementarity and Variational Inequalities in Electronics

70 Complementarity and Variational Inequalities in Electronics

4.2.20 Class of (M, ) ∈ PS0 n
n
We define by PS0 n the set of (M, ) ∈ R n×n × (R ;R ∪{+∞}) such that
n
D( ∞ ) = R (4.18)
and
σ(M) ∩ R ⊂ R + . (4.19)

n
Example 36. If M is a weakly positive semidefinite matrix and ∈ (R ;R ∪
n
{+∞}) with D( ∞ ) = R , then (M, ) ∈ PS0 n .
4.2.21 Class of (M, ) ∈ DS n
n
We define by DS n the set of (M, ) ∈ R n×n × (R ;R∪{+∞}) such that there
n
exists a positive diagonal matrix ∈ R n×n and a subset K ⊂ R of the form
K = K 1 × K 2 × ··· × K n , (4.20)
where, for each i ∈{1,...,n}, K i ⊂ R is a nonempty closed convex cone, such
that

= K (4.21)
and
(∀x ∈ K,x
= 0) : Mx, x > 0. (4.22)
n
Remark 19. If (M, ) ∈ DS n then ( M, ) ∈ PD n with ∈ D (R ;R ∪
{+∞}).
Example 37. Let K = R + × R + × R + , = K , and

⎛ ⎞
1 −20
M = ⎝ 0 1 0 ⎠ .
⎟
⎜
0 2 1
Set
⎛ ⎞
1 0 0
= ⎝ 0 2 0 ⎠ .
⎜
⎟
0 0 1
We have

3
2
2
(∀x ∈ K = R ,x
= 0) : Mx, x = Mx,x = (x 1 − x 2 ) + (x 2 + x 3 ) > 0.
+
Thus (M, ) ∈ DS n .