Mathematics for Economists

Unconstrained local su¢ cient conditions

Lemma

If xTHx is a positive deÖnite quadratic form onRnthen there is anα> 0

such that

xTHxαkxk^2.

Letα>0 be the minimum ofxTHxover the compact setkxk= 1.

Obviously

x

kxk

T

H x

kxk

α> 0

which implies the lemma.