When I was reading a paper related to computer vision, I came across the following notation, where an ellipse is represented by the equation $\mathbf{x}^TM\mathbf{x} = 1$, where the ellipse parameter M, is given by a $2\times2$ matrix where,
$M = \begin{bmatrix}a & b\\ b & c \end{bmatrix}$
I want to know firstly, how does this equation represent an ellipse, and further how does this ellipse gets normalized to a circle with the affine transformation, $\mathbf{x}' = M^{1/2}\mathbf{x}$.
A $2 \times 2$ symmetric matrix $M = \begin{bmatrix} a& b\\ b& c \end{bmatrix}$ defines a quadratic form by $f(x,y) = \begin{bmatrix} x & y \end{bmatrix} M \begin{bmatrix} x \\ y \end{bmatrix} = ax^2 + 2bxy + cy^2$. Thus a conic section (centered at the origin) can be regarded as a $2 \times 2$ symmetric matrix and vice-versa. An ellipse corresponds to the condition $b^2 - 4ac < 0$.
Now, making the coordinate change $\mathbf{x}' =M^{1/2}\mathbf{x}$, we have that $\mathbf{x}^T M \mathbf{x} = \mathbf{x}^T M^{1/2} M^{1/2} \mathbf{x} = (\mathbf{x}')^T I \mathbf{x}'$, so in the new coordinates, the quadratic form looks like, letting $\mathbf{x}' = (x', y')$, $x'^2+y'^2$, which gives the equation for a circle.
EDIT: to get a conic centered at the origin from a quadratic form, set $ax^2+ 2bxy + cy^2 = 1$. Conversely, any non-degenerate conic centered at the origin can be put into such a form by dividing by the constant term.