Eigenvalues of positive definite (p.d) matrix

[roee]

If C = A +B where A,B are both p.d, than C is p.d and its eigenvalues are positive.

Waht can you say about the relationship between the eigenvalues of C, and A,B ?

Thanks.

[mnb96]

Hello,
There is for sure a (non-trivial) relationship but I cannot provide yet a concrete answer, because I myself am working on the same problem at the moment.

I can recall that real symmetric positive definite matrices have a geometrical interpretation: they represent (hyper)ellipsoids in n-space (check the relative lecture at MIT of Gilbert Strang here: http://www.mathvids.com/lesson/mathhelp/349-positive-definite-matrices-tests-tests-for-minimum-and-ellipsoids-in-rn)
The eigenvectors of the matrix represent the directions of the ellipsoid radii, while the corresponding eigenvalues represent their lenghts.

I suspect that studying how addition of two PD matrices affects the geometry of the ellipsoids might give you at least a visual understanding and put you on the right track. I believe that such concepts are used in physics too, but physicists usually refer to 2nd-rank symmetric tensors.

Please if you find a solution to this problem post it here! This is an interesting problem. I also posted some similar questions here in PF, and (quite to my surprise) have never got any answer.

[trambolin]

something useful might be only if they share an eigenvector