QUESTION: Consider the quadratic form $Q : \mathbb{Q}^2 \to \mathbb{R}$ given by
$$Q(x_1, x_2) = 2(x_1 + 2x_2)^2 − (2x_1 − x_2)^2.$$
(a) What is the matrix $A$ associated to this form?
(b) The eigenvalues and eigenvectors of $A$ can be deduced directly from the expression of $Q$ given above. Explain how and state what these eigenvalues and eigenvectors are. No credit will be awarded for using the characteristic equation of $A$ to deduce the eigenvalues.
REMARK: I was able to come up with the matrix $A$ by expanding the quadratic and coming up with $A = -2x_1^2 + 12x_1x_2 + 7x_2^2$. Then expanding that to $A = -2x_1^2 + 6x_1x_2 + 6x_1x_2 + 7x_2^2$.
From there I am lost on how to deduce the eigenvalues and eigenvectors from this equation. I used the characteristic equation to find the eigenvalues are 10, -5 with eigenvectors of $(1, 2)$ and $(-2, 1)$ but how can I can I come to that conclusion without the characteristic equation?
For a diagonal quadratic form, $$Q = \sum_i \alpha_i x_i^2,$$ the eigenvectors of the associated matrix are simply the Euclidean basis vectors, and the eigenvalues are the $\alpha$.
Now your $Q$ does not quite have this form; instead of squaring $x_1$ and $x_2$ it squares linear combinations $x_1+2x_2$ and $2x_1-x_2$. Can you think of a new basis in which $Q$ does become diagonal?