In quantum mechanics, you use a “Hamiltonian matrix” H to encode, in some sense, “everything that a particle in your system is allowed to do” along with some energy associated with that. For instance, an electron in a metallic crystal is allowed to “hop” over to a neighboring atom and that is associated with some kinetic energy. Or it is in some cases allowed to stay on the same atom as another electron, and that is associated with a potential energy (Coulomb repulsion).

The eigenvalues of this matrix is the answer to “what are the energies of each stable electron state in this system”. If you know how many electrons you have (they tend to fill the lowest energy states they can at zero temperature), and you know what temperature you have (which gives you the probability of each “excited” state being occupied), then you can say a lot about the system. For instance, you can say what physical state lowers the “free energy” of the electrons at a given temperature (which can be used to predict phase transitions and spin configurations), or what is the “density of states” (which can be used to predict electronic resistance). You can also obtain the system’s entropy from the eigenvalues alone.

There are however many cases where you might need eigenvectors too, since they usually provide all the spatial information about “where in your system is this stuff happening”. When I need the eigenvectors, CuPy is still hundreds of times faster on my hardware, but the gap is just not as extreme as it was for pure eigenvalue calculation in my benchmarks.