Identify types of quadratic expressions: positive definite, negative definite, indefinite*
Definition
Let f (x, y) be a quadratic form.
- f is said to be positive definite if f (x,y) > 0 for all (x, y) not equal to (0, 0)
- f is said to be positive definite if f (x,y) < 0 for all (x, y) not equal to (0, 0)
-f is said to be indefinite if there exists points (x^+, y^+) and (x^-, y^-) such that f (x^+, y^+) > 0 and f (x^-, y^-) < 0
- f is said to be positive definite if f (x,y) > 0 for all (x, y) not equal to (0, 0)
- f is said to be positive definite if f (x,y) < 0 for all (x, y) not equal to (0, 0)
-f is said to be indefinite if there exists points (x^+, y^+) and (x^-, y^-) such that f (x^+, y^+) > 0 and f (x^-, y^-) < 0
Example.
f (x, y) = x^2 + y^2 is positive definite
f (x, y) = - x^2 - y^2 is negative definite
f (x, y) = x^2 - y^2 is indefinite
f (x, y) = 2xy is indefinite
f (x, y) = - x^2 - y^2 is negative definite
f (x, y) = x^2 - y^2 is indefinite
f (x, y) = 2xy is indefinite