Fig. 5.

The two basic ideas of relative eigenanalysis. (upper left) For any two ellipses, such as covariance matrices of a pair of measurements in two groups, the relative eigenvectors are the directions that are conjugate in both of the ellipses at the same time. (A pair of diameters of an ellipse is conjugate if the tangents at the endpoints of each diagonal are parallel to the other diagonal.) (upper right, lower left) The relative eigenvectors can be computed as well as the axes of either ellipse when the other is linearly transformed into a circle. (For a circle, all pairs of perpendicular diameters are conjugate.) (lower right) Yet another linear transformation of the same pair of ellipses. The natural distance function between two ellipses is the same in all these panels: the square root of the sum of the squares of the logarithms of the ratios of length between the paired diameters (the relative eigenvectors) of the two ellipses. Here that distance is 0.344