Calculating orientation of ellipse from eigen values and eigen vectors

Hi, I have a covariance matrix and i can compute the eigen values and eigen vectors. but from these information, i am unable to formulate an ellipse. I am unable to calculate the orientation of the ellipse.Can any one help me in it and point out any good source of information for calculating the the orientation of ellipse.

Thanks

[Matt J]

Don't the eigenvectors themselves already define the orientation of the ellipse? If not, what does?

[Niraj]

Yes, so correct me if i am wrong. If i need to look at the orientation of an ellipse(angle subtended by the major axis). Then i need to find the eigen vector corresponding to the max eigen value. Then the orientation would be the inverse of the tan of eigen vectors.If V represents the max eigen vector then the orientation would be

atan(V(2,1)/V(1,1))

Right?

Answer 1

[Matt J]

I think the major axis corresponds to the minimum eigenvalue, and the better way to compute the angle would be

atan2(V(2),V(1))

[Niraj]

Hi,Thanks for the reply. But are you sure, we take the min eigen value? I as referring to this pdf slide:

http://cosmic.mse.iastate.edu/library/pdf/pcalevel1.pdf

which says that we take the max eigen value.

[Matt J]

I don't know exactly how you are relating the ellipse to the covariance matrix. However, notice that in this equation for an ellipse

 100*x^2+y^2= 1

the major axis is along y, even though y has the smaller coefficient.

In any case, you should be able to plot the ellipse you think you have generated and verify that it looks as you expect.

[James Tursa]

Major axis goes with max eigenvalue.

[Matt J]

Major axis goes with max eigenvalue.

Without specifying what equations we're talking about, it's just not clear why that would be true. If I assume we're talking about an ellipse equation in the following matrix form

     x.'*A*x<=1

then the major axis corresponds with the minimum eigenvalue, as shown in my last comment, with A=diag([100,1])..

[Niraj]

Thanks for the answers, but Major axis goes with max eigenvalue.It worked for me! Thanks for replies.

[Matt J]

Because the ellipses you are talking about are supposed to be the isocontours of a Gaussian PDF? I can't imagine any other reason.