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

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Homework Help Overview

The discussion revolves around the quadratic form x^2 - xy + y^2 and the inquiry into whether this form consistently maintains the same sign (either positive or negative) across all values of x and y.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the non-negativity of the quadratic form and its behavior under various conditions, such as when |x| < |y|. There are inquiries about the application of Sylvester's criterion and the representation of the quadratic form as a matrix. Questions arise regarding the implications of the determinant being zero and the significance of eigenvalues.

Discussion Status

The discussion is active, with participants offering insights into the properties of the quadratic form and suggesting methods for analysis, such as matrix representation and eigenvalue determination. There is a mix of exploration and questioning of assumptions, particularly regarding the conditions under which the form is defined.

Contextual Notes

Participants are navigating the implications of the quadratic form's behavior, including the conditions for non-negativity and the relevance of matrix representation in the context of eigenvalues and determinants.

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Homework Statement


i have quadratic form
x^2-xy+y^2
how can I check if this form has always same sign (+ or -)?
 
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player1_1_1 said:

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|.
 
i heard about silvester method (with det), how i can solve this with this method?
 
Represent the form as a matrix, M, so x^(T)Mx is your quadratic form. Then find the eigenvalues of M.
 
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?
 
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.
 

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