Principal axis theorem

A Part of Math

The equations in the Cartesian plane

define, respectively, an ellipse and a hyperbola. In each case, the x and y axes are the principal axes. This is easily seen, given that there are no cross-terms involving products xy in either expression. However, the situation is more complicated for equations like

Here some method is required to determine whether this is an ellipse or a hyperbola. The basic observation is that if, by completing the square, the expression can be reduced to a sum of two squares then it defines an ellipse, whereas if it reduces to a difference of two squares then it is the equation of a hyperbola:

Thus, in our example expression, the problem is how to absorb the coefficient of the cross-term 8xy into the functions u and v. Formally, this problem is similar to the problem of matrix diagonalization, where one tries to find a suitable coordinate syst5em in which the matrix of a linear transformation is diagonal. The first step is to find a matrix in which the technique of diagonalization can be applied.

The trick is to write the equation in the following form:

where the cross-term has been split into two equal parts. The matrix A in the above decomposition is a symmetric matrix. In particular, by the spectral theorem, it has real eigenvalues and is diagonalizable by an orthogonal matrix (orthogonally diagonalizable).

To orthogonally diagonalize A, one must first find its eigenvalues, and then find an orthonormal eigenbasis. Calculation reveals that the eigenvalues of A are

with corresponding eigenvectors

Dividing these by their respective lengths yields an orthonormal eigenbasis:

Now the matrix S = [u1 u2] is an orthogonal matrix, since it has orthonormal columns, and A is diagonalized by:

This applies to the present problem of "diagonalizing" the equation through the observation that

Thus, the equation is that of an ellipse, since it is the sum of two squares.