# Principle 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 8 xy** 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* = [**u**1 **u**2] 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.

**To summarize:**

The equation is for an ellipse, since both eigenvalues are positive. (Otherwise, if one were positive and the other negative, it would be a hyperbola.)

The principal axes are the lines spanned by the eigenvectors.

The minimum and maximum distances to the origin can be read off the equation in diagonal form.