These belong to the family of [Euclidean Balls] . In \(R^2\), the ellipsoid is represented by the standard equation \(\dfrac{(x - x_c)^2}{a^2} + \dfrac{(y-y_c)^2}{b^2} \leq 1\). Here \(x_c\) and \(y_c\) are the center coordinates of the ellipse, \(a\) is the semi-major axis and \(b\) is the semi-minor axis. We can represent the standard form equation in the matrix form as
\[\begin{bmatrix} x - x_c & y-y_c \end{bmatrix}^T \begin{bmatrix}a & 0 \\ 0 & b \end{bmatrix}^{-1}\begin{bmatrix}x-x_c \\ y- y_c\end{bmatrix} \leq 1\]The matrix \(\begin{bmatrix}a & 0 \\ 0 & b \end{bmatrix}\) referred to as \(P\) is a [Symmetric Matrix] and also a [Positive Definite Matrix] . It determines how far the ellipsoid extends in each direction from the center of the ellipse. If the L.H.S is less than \(1\) then the points lies inside the ellipse. If the L.H.S. is equal to 1 then the point lies on the boundary of the ellipse. If the L.H.S. is greater than 1 then the point lies outside the boundary of the ellipse.
The length of the semi-axis of the ellipse is given by the \(\sqrt \lambda _i\) where \(\lambda_i\)s are the [Eigen Value]s of the matrix \(P\).
An ellipsoids is a [Convex Set]