Converting two variables quadratic form

Question

I'm currently studying Numerical Linear Algebra, and stumbled upon this question of Matrix Diagonalization.

Given a quadratic form of $q = x_1^2 - 4x_1x_2 + x_2^2$ , convert the equation into quadratic form of $q = ay_1^2 - by_2^2$. Find $a$ and $b$

Here's what I've done so far.

from the initial quadratic form, the matrix expression is

$q = \begin{bmatrix} x_1 & x_2 \end{bmatrix} \begin{bmatrix} 1 & -2 \\ -2 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $

Let $A = \begin{bmatrix} 1 & -2\\ -2 & 1 \end{bmatrix}$

then, by diagonalizing $A$, we get

$A = P DP^{-1}$

with

$P = \begin{bmatrix} -1 & 1 \\ 1 & 1 \end{bmatrix}$ and $D = \begin{bmatrix} 3 & 0 \\ 0 & -1 \end{bmatrix} $

I'm not sure how to continue from this step, any help and insight would be really appreciated, thank you in advance.

Answer 1

Without introducing any matrix notation, an efficient way to solve this problem is "completing squares", as follows: \begin{align*} & x_1^2 - 4x_1x_2 + x_2^2 \\ = & x_1^2 - 4x_2x_1 + 4x_2^2 - 3x_2^2 \\ = & (x_1 - 2x_2)^2 - (\sqrt{3}x_2)^2. \end{align*} Therefore $y_1 = x_1 - 2x_2, y_2 = \sqrt{3}x_2$ is the desired transformation, with $a =1, b = 1$.