Mathematics for Economists

Constrained positive deÖnite matrixes

After some calculation one gets that:

(^1) Write the basis vectors ofBÖrst.B=(B 1 ,B 2 )=(B 1 ,B 1 U),where
B 1 forms a basis for the column space ofBandUare the coordinates
of the other columns inB 2.
(^2) Form the bordered Hessian
0
BB
@

0

pp

B

(pn)

BT

(np)

A

(nn)

1

CC

A

(^3) Check the determinants of the lastnpleading principal minors.
(^4) The sign of these determinants must be( 1 )pin the constrained
positive deÖnite(minimum) case. In the negative deÖnite (maximum)
case the signs must alter staring with( 1 )nfor the largest minor.