Mathematics for Economists

Constrained positive deÖnite matrixes

Problem

LetAbe symmetric. When does xTAx 0 for everyBx= 0 ?Or when

does xTAx> 0 for everyBx= 0 ,x 6 = 0?

We can assume thatBis fat and full rank. Solving the homogeneous

equationx 2 =Dx 1 wherex 1 is the vector of free variables. If the

number of the equations ispthen the dimension ofx 1 isnp. Hence

ker(B)=



x=Pyjy 2 Rnp

=



x 1

x 2



=



I

D



yjy 2 Rnp



.

Hence one should study

xTAx=yTPTAPy$yTCy, dim



PTAP



=dim(C)=np

assize(P)=n(np).