Mathematics for Economists

Constrained positive deÖnite matrixes

Example

Solve the problemx 1 x 2 x 3 !max,x 12 +x 22 +x 32 = 1

The Lagrangianx 1 x 2 x 3 +λ

x 12 +x 22 +x 32  1

.The Örst order conditions

x 1 x 2

2 x 3

=x^2 x^3

2 x 1

=x^1 x^3

2 x 2

=λ,

Obviously the solutions of the Örst order conditions are

x 12 = x 22 =x 32 =

1

3.

A =

0

@

2 λ x 3 x 2

x 3 2 λ x 1

x 2 x 1 2 λ

1

A,B= 2 x 1 2 x 2 2 x 1 
.