I'm quite confused with all the ssh security stuff. I am trying to reconfigure a system that is currently broken for reasons unknown. Machine A is your personal computer that you use whenever you're at home. Machine B is the head node of an HPC cluster and all the other machines C are all identically configured machines which share the home directories of machine B. This is an HPC cluster if you haven't guessed. How would I configure passwordless ssh between any nodes B or C. A can only get to C through sshing into B
Not sure whether you want to specify that it should only work if you come from A?
grawity's answer is probably better wrt security, but here's an answer for speed of setup and/or working from multiple A's..
Going on your whole... "machines C are all identically configured machines which share the home directories of machine B"
This'll mean anyone on B or C... (or any machine with access to that home directory's .ssh/) will be able to ssh between them at will, wheras I think grawity's requires you to have come from A, and have set up that machine appropriately.