Page 73 - Complementarity and Variational Inequalities in Electronics

64 Complementarity and Variational Inequalities in Electronics

For example,
⎛ ⎞
3 −2 −1
M = ⎝ 0 −1 −2 ⎠
⎜
⎟
−1 −3 0
is a Z-matrix. Note that any Z-matrix M can be written as

M = αI − P
with α ∈ R and P ≥ 0 (in the sense that ∀i,j ∈{1,...,n}: p ij ≥ 0). For exam-
ple, the matrix M can be written as

⎛ ⎞ ⎛ ⎞
100 021
M = 3 ⎝ 010 ⎠ − ⎝ 042 ⎠ .
⎟
⎜
⎜
⎟
001 133
4.2.10 M-Matrix
We say that M is an M-matrix if

M = αI − P

for some P ≥ 0 and some α> ρ(P) with ρ(P) denoting the spectral radius
of P , that is,
ρ(P) = max{|λ|: λ ∈ σ(P)}.

The following theorem gives some important relations between different classes
of Z-matrices (see e.g. [54]).

Theorem 3. Let M ∈ R n×n be a Z-matrix. Then the following conditions are
equivalent:
1. M is an M-matrix.
2. M is positive stable.
3. M is nonsingular, and M −1 ≥ 0.
T
4. There exists a positive diagonal matrix D such that DA + A D is positive
definite.

For example, the matrix
⎛ ⎞
1 −1 −1
M = ⎝ 0 2 −1 ⎠
⎜
⎟
0 0 3