Page 68 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 59

4.2.2 Positive Definite Matrix

We say that M is positive definite if
n
(∀x ∈ R ,x
= 0) : Mx,x > 0.
Note that M ∈ R n×n is positive definite if and only if the symmetric matrix
T
M + M is positive definite since
1 T
Mx,x = (M + M )x,x .
2
Sylvester’s criterion ensures that M is positive definite if and only if all of the
T
leading principal minors of M + M are positive, that is,
T
(∀1 ≤ k ≤ n) : 12...k (M + M )> 0.
For example, the matrix

⎛ ⎞
2 −10
M = ⎝ 0 2 0 ⎠
⎟
⎜
0 0 1
is positive definite. Indeed,
⎛ ⎞
4 −10
T ⎜ ⎟
M + M = ⎝ −1 4 0 ⎠ ,
0 0 2

T
T
T
and 1 (M + M ) = 4, 12 (M + M ) = 15, and 123 (M + M ) = 30.
It is also known that M is positive definite if and only if all of the eigenvalues
T
of M + M are positive, that is,
T
(∀λ ∈ σ(M + M )) : λ> 0.
For example, the matrix
⎛ ⎞
200
⎜ ⎟
M = ⎝ 120 ⎠
004
is positive definite. Indeed,
⎛ ⎞
4 1 0
T ⎜ ⎟
M + M = ⎝ 1 4 0 ⎠
0 0 8