Mathematics for Economists

Constrained positive deÖnite matrixes

Example

Solvex 12 +x 22 !min,x 1 +x 2 = 1.

The Lagrangian isL(x 1 ,x 2 ,λ)=x 12 +x 22 +λ(x 1 +x 2  1 ).

The necessary condition

2 x 1 +λ = 0 , 2 x 2 +λ= 0 )x 1 =x 2 ,)x 1 =x 2 =^1

2

.

A =



2 0

0 2



,B=

1 1

,H=

0

@

0 1 1

1 2 0

1 0 2

1

A

det(H)= 4 .One should check 21 determinants and for minimum the

sign must be 1 =( 1 )pas there is just one constraint,p=1. Hence

we have a local minimum.