We are given the equation $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ We did an example of this in class but the equation had less terms. I took a note in class that says : if there are linear terms, I have ...
The matrix $M=AVA^T - BCB^T +D$ is known to be positive semidefinite (PSD), where $V, C, D$ are each diagonal matrices with positive values, and $V, C$ has small size when compared to the size of $M$. ...
Where an ellipse is expressed in quadratic form (e.g. $ax^2 + bxy + cy^2 = k$ is expressed as $x^TQx = k$), the principal axes are in the directions given by the eigenvectors of Q. I understand this. ...
I'm having trouble with some maths regarding the expression of the matrix quadratic form (i.e. $x^TAx$) and the proof that, where the eigenvalues $\lambda_1,\lambda_2,...,\lambda_n$ are all positive, ...