Page 71 - Complementarity and Variational Inequalities in Electronics

62 Complementarity and Variational Inequalities in Electronics

For example, the matrix

⎛ ⎞
1 −2 −8
M = ⎝ 0 1 2 ⎠
⎟
⎜
0 0 2
is a P-matrix. Indeed, we have 1 (M) = 1, 2 (M) = 1, 3 (M) = 2,
12 (M) = 1, 13 (M) = 2, 23 (M) = 2, and 123 (M) = 2.

4.2.7 P0-Matrix

We say that M is a P0-matrix if all the principal minors of order k of M are
nonnegative, that is,

≥ 0.
(∀1 ≤ k ≤ n) : (M) i 1 i 2 ...i k
It is known that M is a P0-matrix if and only if
n
(∀x ∈ R ,x
= 0)(∃ α ∈{1,...,n}) : x α
= 0& x α (Mx) α ≥ 0.
For example, the matrix

⎛ ⎞
1 −2 −8
M = ⎝ 0 0 2 ⎠
⎜
⎟
0 0 4
is a P0-matrix. Indeed, we have 1 (M) = 1, 2 (M) = 0, 3 (M) = 4,
12 (M) = 0, 13 (M) = 4, 23 (M) = 0, and 123 (M) = 0.

4.2.8 Positive Stable Matrix
We say that M is positive stable if i + (M) = n, that is,

(∀λ ∈ σ(A)) : re(λ) > 0.

For example, the matrix
⎛ ⎞
1 −10
⎟
M = ⎝ 1 1 0 ⎠ (4.6)
⎜
0 0 1
is positive stable since σ(M) ={1 − i,1 + i,1}.
If M is positive definite, then M is also positive stable. A more general
relation between positive stable and positive definite matrices is given by Lya-
punov’s theorem (see e.g. [54]).