10000 by 10000 isn't really very big. A double precision matrix of this size requires 800 megabytes of memory. You'll need at least as much memory to hold the matrix of resulting eigenvectors, and you'll need additional working storage, but this is well within the capability of a typical desktop machine with 8 gigabytes or more of memory. It would also fit into the graphics memory on a higher end GPGPU.
In terms of time, this also isn't too bad. I just computed the eigenvalues and eigenvectors for a random symmetric 100000 by 10000 matrix on a desktop machine (using MATLAB/LAPACK and running with 8 cores), and it took about 3 minutes to run.
If you only need to do this computation for a small number of matrices of this size, then it's likely not worth your time and effort to move this to a GPGPU. However, if you plan on doing these computations 24/7, then putting some effort into speeding them up might be appropriate.
This is the sort of computation that you might be able to speed up with a high end GPGPU, but don't be surprised if it doesn't perform that well on a more basic graphics card. The issues here are that your graphics card might not have sufficient memory, and that double precision performance is not a strong point of many GPGPU's.
Since eigenvalue computation is commonly available through the LAPACK libraries, I'd suggest that you look for an implementation of the LAPACK libraries that has been optimized for your GPGPU. In particular, check out the OpenCL version of the MAGMA library: