Mathematics for Economists

Constrained positive deÖnite matrixes

Example

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

The Lagrangian is

L(x 1 ,x 2 ,x 3 ,λ)=x 12 +x 22 +x 32 +λ(x 1 +x 2 +x 3  1 ).

Obviouslyx 1 =x 2 =x 3 = 1 /3 is the solution.

A=

0

@

2 0 0

0 2 0

0 0 2

1

A,B=1 1 1
,H=

0

BB

@

0 1 1 1

1 2 0 0

1 0 2 0

1 0 0 2

1

CC

A

The determinant ofHis 12.