Mathematics for Economists

Constrained positive deÖnite matrixes

The Hessian of the Lagrangian is

A=



2 λ 1

1 2λ



,B=( 2 x 1 , 2 x 2 ).

For the Örst root



p^1

2 ,

p^1

2



,λ= 1 / 2

H=

0

@

0 2/

p

2 2/

p

2

2 /

p

2  1 1

2 /

p

2 1  1

1

A

det (H)=8 so asn=2 the sign for maximum is( 1 )n= 1 ,hence

p^1

2 ,

p^1

2



is a local maximum.