Eigenvalues, eigenvectors and error ellipses
An alternative approach to finding the parameters of error ellipses is to use eigenvalues and eigenvectors. The eigenvalues of the matrix defined by the relevant parts of the variance-covariance of parameters give the squares of the sizes of the semi-major and semi-minor axes of the error ellipse. The eigenvectors give their directions.
The eigenvalue problem is to find values of λ and z that satisfy the equation:
where N−1 is defined by the relevant parts of the variance-covariance of parameters.
A control network with absolute and relative error ellipses
The characteristic polynomial is derived from the determinant of the sub-matrix above:
so that the two solutions for λ are found from the quadratic equation:
The solutions for this quadratic equation are:
The magnitudes of the semi-major and semi-minor axes are the square roots of the two solutions for λ. The directions of the axes are found from the eigenvector associated with each eigenvalue. The eigenvector associated with λ1 is found from:
so that the direction of the axis associated with λ1 is given by:
Both formulae give the same answer. λ2 is found in the same way and will be exactly 90◦ different from λ1