Page 69 - Complementarity and Variational Inequalities in Electronics

60 Complementarity and Variational Inequalities in Electronics

and
T
σ(M + M ) ={3,5,8}.

4.2.3 Positive Semidefinite Matrix
We say that M is positive semidefinite if:

n
(∀x ∈ R ) : Mx,x ≥ 0.
Sylvester’s criterion ensures that M is positive semidefinite if and only if all the
T
principal minors of M + M are nonnegative, that is,
T
(M + M ) ≥ 0.
(∀1 ≤ k ≤ n) : i 1 i 2 ...i k
For example, the matrix
⎛ ⎞
4 −10
M = ⎝ −2 1 0 ⎠
⎟
⎜
0 0 0
is positive semidefinite. Indeed, we have
⎛ ⎞
8 −30
T
M + M = ⎝ −3 2 0 ⎠
⎜
⎟
0 0 0
T T T
and 1 (M + M ) = 8, 2 (M + M ) = 2, 3 (M + M ) = 0, 12 (M +
T
T
T
T
M ) = 7, 13 (M + M ) = 0, 23 (M + M ) = 0, and 123 (M + M ) = 0.
It is also known that M is positive semidefinite if and only if all the eigen-
T
values of M + M are nonnegative, that is,
T
(∀λ ∈ σ(M + M )) : λ ≥ 0.
For example, the matrix
⎛ ⎞
4 −10
M = ⎝ −1 4 0 ⎠
⎜
⎟
0 0 0
is positive semidefinite. Indeed, we have
⎛ ⎞
8 −20
T ⎜ ⎟
M + M = ⎝ −2 8 0 ⎠
0 0 0