How to bring elliptical equation $2x^2+2y^2+3xy-x-y=0$ into canonical form I have this ellipse: $$2x^2+2y^2+3xy-x-y=0$$
Canonical form is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
How can I bring my ellipse to that canonical form? It seems like I need some substitution.

The center of the ellipse is where the gradient vanishes,
$$4x+3y-1=0,\\4y+3x-1=0.$$
Then we center around the solution $\left(\dfrac17,\dfrac17\right)$,
$$2\left(x+\frac17\right)^2+2\left(y+\frac17\right)^2+3\left(x+\frac17\right)\left(y+\frac17\right)-\left(x+\frac17\right)-\left(y+\frac17\right)
\\=2x^2+3xy+2y^2-\frac17=0.$$
In matrix form,
$$z^TAz-\frac17=0$$ where $A=\begin{pmatrix}2&\frac32\\\frac32&2\end{pmatrix}$.
We diagonalize the matrix and find the Eigenvalues $\dfrac12$ and $\dfrac72$. Hence the reduced equation
$$\frac{x^2}2+\frac{7y^2}2=\frac17.$$
