Page 70 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 61

and

σ(A) ={0,6,10}.

4.2.4 Strictly Copositive Matrix

We say that M is strictly copositive if

(∀x ≥ 0,x
= 0) : Mx,x > 0.

For example, the matrix
⎛ ⎞
110
⎜ ⎟
M = ⎝ 110 ⎠
001
2
2
is strictly copositive. Indeed, we have Mx,x = (x 1 + x 2 ) + x , and it is clear
3
that if x ≥ 0,x
= 0, then Mx,x > 0. The matrix is however not positive defi-
nite.
4.2.5 Copositive Matrix

We say that M is copositive if

(∀x ≥ 0) : Mx,x ≥ 0.

For example, the matrix
⎛ ⎞
120
⎜ ⎟
M = ⎝ 210 ⎠
000
is copositive. The matrix is however neither strictly copositive nor positive
semidefinite.

4.2.6 P-Matrix

One says that M is a P-matrix if all principal minors of order k of M are positive,
that is,

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