Page 65 - Complementarity and Variational Inequalities in Electronics

56 Complementarity and Variational Inequalities in Electronics

Example 23 (Practical Zener diode model). The recession function of the elec-
trical superpotential of the practical Zener diode (see Section 2.3.4 in Chapter 2)
is
(∀x ∈ R) : (ϕ Z ) ∞ (x) = {0} (x).

4.2 SPECIAL CLASSES OF MATRICES M AND FUNCTIONS

In the study of Problem VI(M,q, ), the following definitions of various spe-
cial matrices M and functions will be used. Let M ∈ R n×n , and let U (1) , U (2) ,
n
..., U (p) ∈ R . We denote by vect{U (1) ,U (2) ,...,U (p) } the set of all possible
linear combinations of the vectors U (1) , ..., U (p) , that is,
p
vect{U (1) ,U (2) ,...,U (p) }={ α i U (i) : α 1 ,α 2 ,...,α p ∈ R}.
i=1
We denote by R(M) the range of M and by ker(M) the kernel of M. We recall
that
R(M) = vect{M (1) ,M (2) ,...,M (n) },

where M (1) ,M (2) ,...,M (n) denote the columns of M.Wealsohave
T
R(M ) = ker(M) ⊥
and
T
⊥
ker(M ) = R(M) .
Moreover,
n T T
R = ker(M) ⊕ R(M ) = R(M) ⊕ ker(M ).
We denote by p M the characteristic polynomial of M, that is,

(∀λ ∈ C) : p M (λ) = det(λI − M).

Let σ(M) ⊂ C be the set of eigenvalues of M, that is,
n
σ(M) ={λ ∈ C :∃ U ∈ C \{0}: MU = λU}.
It is well known that

σ(M) ={λ ∈ C : p M (λ) = 0}.

For k ∈ N, we denote by M i 1 i 2 ...i k the principal submatrix of M, which is
i 1 i 2 ...i k
obtained by choosing the lines and columns whose indexes i 1 <i 2 < ··· <i k