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Unformatted text preview: 3.8. LOCAL EXTREMA 359 Definite Quadratic Forms Since it is homogeneous, every quadratic form is zero at the origin. We call the quadratic form Q definite if it is nonzero everywhere else: Q ( −→ x ) negationslash = 0 for −→ x negationslash = −→ 0 . For example, the forms Q ( x,y ) = x 2 + y 2 and Q ( x,y ) = − x 2 − 2 y 2 are definite, while Q ( x,y ) = x 2 − y 2 and Q ( x,y ) = xy are not. We shall see that the form Q ( x,y ) = 2( x + y ) 2 + y ( y − 6 x ) = 2 x 2 − 2 xy + 3 y 2 is definite, but a priori this is not entirely obvious. A definite quadratic form cannot switch sign, since along any path where the endpoint values of Q have opposite sign there would be a point where Q = 0, and such a path could be picked to avoid the origin, giving a point other than the origin where Q = 0: Remark 3.8.1. If Q ( −→ x ) is a definite quadratic form, then one of the following inequalities holds: • Q ( −→ x ) > for all −→ x negationslash = −→ ( Q is positive definite...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
- Fall '08