I have to deal with very large matrices in Mathematica (dimensions $10^4\times10^4$ at least). Obtaining the eigenvalues of these matrices is not so difficult since, it is not memory intensive or particularly slow. Finding the eigenvectors is not slow either, but it is particularly memory intensive, since Mathematica first finds and stores all the eigenvectors and then allows for manipulations using them.
What I would like to do is compute a function that depends on the eigenvectors and eigenvalues, but without dealing with the memory requirements.
Is there any good way within Mathematica where I can compute the eigenvectors one by one , use them to compute the function and then throw them away? I was unable to spot anything within the documentation, but if there is something in there that I missed, it would be nice if someone could point it out.