How do you determine whether the quadratic form is positive and negative definite?

Question

How do you determine whether the quadratic form $Q(x,y) = 2x^2 - 4xy + 5y^2$ is positive definite, negative definite, or indefinite?

Could someone show step by step with explanations? Thank you

Answer 1

Diagonalize. In this case, it comes down to completing the square.

Answer 2

If I remember correctly, it is equivalent to check the corresponding property of its Hessian matrix ($2\times 2$ symmetric matrix in this case), e.g. by looking at its eigenvalues.

Answer 3

By computing the bilinear form $B((x_1,y_1),(x_2,y_2))=\frac{1}{2}[Q((x_1,y_1)+(x_2,y_2))-Q((x_1,y_1))-Q((x_2,y_2))]$

and testing it with the points $(1,0)$ and $(0,1)$, you can discover that this is the matrix for the form is

$$ \begin{bmatrix}2&-2\\-2&5\end{bmatrix} $$

This matrix has eigenvalues 1 and 6, so it is positive definite.

Answer 4

We can use the method of gauss to reduce the quadratic form: $$Q(x,y) = 2x^2 - 4xy + 5y^2=2(x-y)^2+3y^2$$ hence the signature is $(2,0)$ and the quadratic form is positive definite.

Answer 5

Compute its discriminant (in this case $\Delta = (-4)^2 - 4 \cdot 2 \cdot 5 = -24$). If it's positive then the form is indefinite, if it's zero then the form is semi-definite, if it is negative the form is definite. Positive or negative (semi-)definite can be seen from either coefficient of $x^2$ or $y^2$.

This is a special case of Sylvester's criterion.