Page 95 - Complementarity and Variational Inequalities in Electronics

86 Complementarity and Variational Inequalities in Electronics

3
The matrix M is strictly copositive, and D( ) ∞ = R . Thus (M, ) ∈ PD n
+
3
and R(M, ) = R .Wehave
⎛ ⎞
1
ker(M) = vect{⎝ −1 ⎠}
⎟
⎜
0
and
3
R ∩ ker(M) ={0}.
+
Example 44. Let
⎛ ⎞
1 −2 −8
M = ⎝ 0 1 2 ⎠
⎜
⎟
0 0 2
and
3
(∀x ∈ R ) : (x) = 3 (x).
R
+
3 j j
The matrix M is a P-matrix and x ∈ D( ) ∞ = R ; therefore x,e e ∈
+
3
3
D(( ∞ ) = R (1 ≤ j ≤ n). Thus (M, ) ∈ P n and R(M, ) = R .Wealso
+
note that M is nonsingular, and thus
3
R ∩ ker(M) ={0}.
+
Example 45. Let
⎛ ⎞
−1 −10
M = ⎝ 1 −10 ⎠
⎟
⎜
0 0 1
and
3
(∀x ∈ R ) : (x) =|x 1 |+|x 3 |.
Here σ(M) ={−1 − i,−1 + i,1}, and thus σ(M) ∩ R ⊂]0,+∞[.Wealsohave
3
∞ = , and thus D( ∞ ) = R. Thus (M, ) ∈ PS n and R(M, ) = R .Note
that the matrix M is nonsingular and thus

D( ∞ ) ∩ ker(M) = ker(M) ={0}.

4.7 NONNEGATIVITY AND SOLVABILITY CONDITIONS
n
Let : R → R ∪{+∞} be a proper convex lower semicontinuous function
with closed domain, and let M ∈ R n×n . In this section, we will assume that
(M, ) ∈ Q0 n .