Generalised Equations
More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the equation
where A is a positive definite matrix and x, v are vectors.
The eigenvectors of A define the principal directions of the ellipsoid and the eigenvalues of A are the squares of the semi-axes:, and . An invertible linear transformation applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable rotation, a consequence of the polar decomposition (also, see spectral theorem). If the linear transformation is represented by a symmetric 3-by-3 matrix, then the eigenvectors of the matrix are orthogonal (due to the spectral theorem) and represent the directions of the axes of the ellipsoid: the lengths of the semiaxes are given by the eigenvalues. The singular value decomposition and polar decomposition are matrix decompositions closely related to these geometric observations.
Read more about this topic: Ellipsoid