Quadratic Form x^2-xy+y^2: Determining Sign Consistency

[player1_1_1]

Homework Statement

i have quadratic form
$$x^2-xy+y^2$$
how can I check if this form has always same sign (+ or -)?

 

[Tinyboss]

Homework Statement

i have quadratic form
$$x^2-xy+y^2$$
how can I check if this form has always same sign (+ or -)?

It's non-negative for all x,y, and 0 only at 0,0. Suppose |x|<|y|. Then |xy|<y^2, and the expression is positive. Likewise if |y|<|x|.

 

[player1_1_1]

i heard about silvester method (with det), how i can solve this with this method?

 

[Dick]

Represent the form as a matrix, M, so x^(T)Mx is your quadratic form. Then find the eigenvalues of M.

 

[player1_1_1]

well, I know that:) but why this form is represented by this matrix? why this form is defined when its more than 0 and no defined when less? and what can I do when det is 0?

 

[Dick]

I think the answers would be a lot clearer if you actually tried to do the problem. What is the matrix M and what are it's eigenvalues? It only has two and they are both positive.