Mathematics for Economists

Constrained positive deÖnite matrixes

Example

Solvex 12 x 22 x 32 !max,x 12 +x 22 +x 32 = 1.

The Lagrangian

x 12 x 22 x 32 +λ

x 12 +x 22 +x 32  1

.

The necessary condition is

2 x 1 +λ 2 x 1 = 2 x 1 ( 1 +λ)= 0

 2 x 2 +λ 2 x 2 = 2 x 2 ( 1 +λ)= 0

 2 x 3 + 2 λx 3 = 2 x 3 ( 1 +λ)= 0.

The relevant solution for the maximum isx 1 = 1 ,x 2 =x 3 = 0.