Numerical Methods and Optimization

Preliminaries 23

where a

ij

,i,j=1,...,n are the coe

ffi

cients o

f

the quadratic

f

orm, x =

[x

1

,...,x

n

]

T

is the vector of variables.

Let A =[a

i,j

]

n×n

bethematrixofcoefficientsofQ(x). Then Q(x)=x

T

Ax =

x

T

Qx,whereQ =(A+A

T

)/2 is a symmetric matrix. Therefore, any quadratic

form is associated with the symmetric matrix of its coefficients.

Definition 1.33 (Definite (Semidefinite) Matrix) A symmetric ma-

trix Q and the corresponding quadratic form q(x)=x

T

Qx is called

• positive definite if q(x)=x

T

Qx > 0 for all nonzero x ∈ IR

n

;

• positive semidefinite if q(x)=x

T

Qx ≥ 0 for all x ∈ IR

n

;

• negative definite if q(x)=x

T

Qx < 0 for all nonzero x ∈ IR

n

;

• negative semidefinite if q(x)=x