How do you determine whether the quadratic form is positive and negative definite?
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
$\endgroup$5 Answers
$\begingroup$Diagonalize. In this case, it comes down to completing the square.
$\endgroup$ 2 $\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ 1 $\begingroup$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.
$\endgroup$ $\begingroup$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.
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