If I draw the curve represented by equation $$x^2+2xy-y^2=1$$, I see a rotated hyperbola. How to transform this equation in oder to obtain the standard form $$X^2-Y^2=1$$?
Using the hint provided by the first answer, I got eigenvalues $$\pm\sqrt{2}$$ and eigenvectors: $$\begin{bmatrix} 1&1\\\frac{1}{1-\sqrt{2}}&\frac{1}{1+\sqrt{2}} \end{bmatrix}$$
How to determine how much rotation needs to occur?
HINT: Diagonalize the matrix $$\begin{bmatrix} 1&1\\1&-1 \end{bmatrix}.$$