Let's say I have a equation
$$X_1^T \Omega X_1 =1 $$
$X_1$ is a $2\times 1$ matrix. $\Omega$ is a $2\times 2$ matrix. This defines an ellipse. $\Omega$ is a positive, semi definite, symmetric matrix. I have the following questions:
1. Is the ellipse defined by $\Omega$ axis aligned or non aligned or we cannot say?
2. If $\Omega$ only contains non zero elements in the diagonal is it definitely axis aligned?
3. If $\Omega$ contains non zero elements on and off the diagonal (still being symmetric) is it non-axis aligned?
4. If an ellipse is non axis aligned whats the link with semi definite programming?