Page 77 - Complementarity and Variational Inequalities in Electronics

68 Complementarity and Variational Inequalities in Electronics

4.2.15 Class of (M, ) ∈ PD n
n
We define by PD n the set of (M, ) ∈ R n×n × (R ;R ∪{+∞}) such that
(∀x ∈ D( ) ∞ ,x
= 0) : Mx,x > 0. (4.10)

Example 27. If M is positive definite, then condition (4.10) is obviously satis-
fied.
Example 28. If M is strictly copositive, then (M, R ) ∈ PD n .
n
+
Example 29. Let
⎛ ⎞
1 1 0
M = ⎝ 1 1 0 ⎠
⎟
⎜
0 0 1
and
3
(∀x ∈ R ) : (x) = K (x)
with
3
K ={x ∈ R : x 1 ≥ 1,x 2 ≥ 1,x 3 ≥ 1}.
3
The matrix M is strictly copositive, and D( ) ∞ = R . Thus (M, ) ∈ PD n .
+
4.2.16 Class of (M, ) ∈ PD0 n
n
We define by PD0 n the set of (M, ) ∈ R n×n × (R ;R ∪{+∞}) such that
(∀x ∈ D( ) ∞ ) : Mx,x ≥ 0. (4.11)
Example 30. If M is positive semidefinite, then condition (4.11) is obviously
satisfied.
Example 31. If M is copositive, then (M, R ) ∈ PD0 n .
n
+
Example 32. Let

1 1
M =
0 1
and
3
(∀x ∈ R ) : (x) = K (x)
with
2
K ={(x 1 ,x 2 ) ∈ R : x 1 ≥ 0,x 1 x 2 ≥ 1}.
2
The matrix M is copositive, and D( ) ∞ = R . Thus (M, ) ∈ PD0 n .
+