Diagonal Quadric Form

Question

I have the equation: $Q(x,y) = 5x^2 - 6xy + 5y^2$. The first question is to write this in $ Q(z) = z^{T}Az$, in Matrixnotation.

The matrix I calculated is: $$\begin{bmatrix} 5 & -3 \\ -3 & 5 \end{bmatrix} $$

Then I have to diagonalize it, such that $A = UDU^{T}$ in order to use another Basis (from $z \to \overline{z}$).

[EDIT] I solved this part using the diagonalization, and obtaining $$\begin{bmatrix} 2 & 0 \\ 0 & 8 \end{bmatrix} $$ as the matrix for the new quadric. Now how can I solve the next question?

At the end I have to draw the new curve given by $z^{T}Az = 8$.

I tried to read some documentation but I don't understand how to procede, can someone help? Thanks!

Answer 1

$$\begin{pmatrix}5&-3\\-3&\;\;5\end{pmatrix}\stackrel{R_2+\frac35R_1}\rightarrow\begin{pmatrix}5&-3\\0&\frac{16}5\end{pmatrix}\stackrel{C_2+\frac35C_1}\rightarrow\begin{pmatrix}5&0\\0&\frac{16}5\end{pmatrix}$$

and there you are...