The two covariance matrices C1 (blue standard ellipse) and C2 (red standard ellipse) of a 2-vector of parameters can be specified by the semi axes of their standard ellipses x' C1-1 x=1 and x' C2-1 x=1, namely (a1,b1) for C1 and (a2,b2) for C2 using the sliders in the left upper corner. The direction of the major axes can be changed directly by clicking on the major axes and moving the cursor.
In case one treats the covariance C1 as reference covariance matrix, which represents the desired structure of the precision, one can require that the covariance matrix C2, derived from a certain measuring design, fully lies in C1. This is equivalent to require, that all functions of the parameters x are more precise when calculated with C2 than when calculated with C1 using variance propagation.
This requirement leads to determining the maximum of the so-called Rayleigh coefficient λ = d' C2 d/ (d' C1 d). It is obtained as largest eigenvalue of the generalized eigenvalue problem |C2 -λ C1| =0.
The applet determines the square roots of the generalized eigenvalues and shows them as √λ1 and √λ2 in the left lower corner.
The eigenvalues can be easily be characterized geometrically. Each ray from the centre intersects the two ellipses. For two such directions, except for symmetry, the tangents at these intersection points at the two ellipses are parallel, here in A1/B1, and in A'1/B'1 for C1 and in A2/B2, and in A'2/B'2 for C2, see the animation Pedal Curve. The eigenvectors are perpendicular to these tangents. However, the eigenvectors in general are not mutually orthogonal.
- Change all three parameters of for C2 (red), namely the two semi axes and the direction, such that the maximal eigenvalue becomes 1. Is this always possible?
- Choose the parameters of the covariance matrix for C2 (red) such that both eigenvalues are 0.8 (alternatively 1.5).
- Under which conditions are the eigenvectors mutually orthogonal?
- Under which conditions is problem 1 solvable?