Answer
See below
Work Step by Step
1. Find eigenvalues:
(A-$\lambda$I)$\vec{V}$=$\vec{0}$
$\begin{bmatrix}
1-\lambda & 3\\
3 & 1-\lambda
\end{bmatrix}\begin{bmatrix}
v_1\\
v_2
\end{bmatrix}=\begin{bmatrix}
0\\
0
\end{bmatrix}$
$\begin{bmatrix}
1-\lambda & 3\\
3 & 1-\lambda
\end{bmatrix}=0$
$(1-\lambda)(1-\lambda)-9=0$
$\lambda_1=4, \lambda_2=-2$
From that we can find eigenvectors: $\{(1,1);(-1,1)\}$
Hence, the set of principal axes for the given quadratic form is: $\{\frac{1}{\sqrt 2}(1,1);\frac{1}{\sqrt 2}(-1,1)$
The sum of squares now can be $4y^2_1-2y^2_2$
