How Can I Derive a Covariance Matrix from Known Eigenvalues and an Eigenvector?

[Lindley]

I have a Gaussian distribution. I know the variance in the directions of the first and second eigenvectors (the directions of maximum and minimum radius of the corresponding ellipse at any fixed mahalnobis distance), and the direction of the first eigenvector.

Is there a simple closed form equation to derive the corresponding covariance matrix? It seems like there should be, since if H0 is the first eigenvector and H1 is the second, then

H0*P*H0^T = var0
H1*P*H1^T = var1

However, this is only two equations and there are 3 unknowns (Pxx, Pxy, and Pyy). Any help?

 

[lanedance]

I don't 100% understand you formulation... however i read as following:
- the problem is 2D guassian distribution

you want to find the 2x2 covariance matrix,
$$P = \begin{pmatrix} P_{xx} & P_{xy} \\ P_{yx} & P_{yy} \end{pmatrix}$$

which is a symmetric (Pxy = Pyx) positive semi-definite matrix, you know the eigenvalues, and an eigenvector of the matrix

in fact as it it symmetric, eigenvectors will be orthogonal, you know the direction and of both eigenvectors (call them v1, v2)

why not use the eigenvectors to create a matrix to diagonalise, call it V, then
[tex]V = [v_1, v_2] [/tx]<br /> $$V^T P V = D$$<br /> <br /> where D is a diagonal matrix with eignevalues on the diagonal, then <br /> $$P = V D V^T$$[/tex]