Principle Axis Theorem
A Part Of Math
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).
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.
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.