Page 75 - Complementarity and Variational Inequalities in Electronics

66 Complementarity and Variational Inequalities in Electronics

The matrices G and H are symmetric, and thus

T
Hx,x C = Ha,a + Hb,b + i Ha − H a,b = Ha,a + Hb,b
n
and

T
Gx,x C = Ga,a + Gb,b + i Ga − G a,b = Ga,a + Gb,b .
n
We have

Ha,a + Hb,b + 2i Sa,b = 2λ( Ga,a + Gb,b ),

and thus

Ha,a + Hb,b = 2re(λ)( Ga,a + Gb,b ).
We have Ha,a + Hb,b ≥ 0 and Ga,a + Gb,b > 0, so that re(λ) ≥ 0.

4.2.12 Weakly Positive Definite Matrix
We say that M is weakly positive definite if all real eigenvalues of M are posi-
tive, that is,
σ(M) ∩ R ⊂]0,+∞[.

For example, the matrix
⎛ ⎞
0 0 −2
M = ⎝ 1 2 1 ⎠
⎜
⎟
1 0 3
is weakly positive definite since σ(M) ={1,2}. For another example, the matrix

⎛ ⎞
−1 −10
M = ⎝ 1 −10 ⎠
⎟
⎜
0 0 1
is weakly positive semidefinite since σ(M) ={−1 − i,−1 + i,1}.

4.2.13 Weakly Positive Semidefinite Matrix
We say that M is weakly positive semidefinite if all real eigenvalues of M are
nonnegative, that is,
σ(M) ∩ R ⊂[0,+∞[.