Doing some quantum mechanics, I just ran into the notion of a matrix whose elements have matrix values for the first time. Specifically, a 2x2 matrix whose elements are 4x4 matrices. This got me wondering how I can extend the question into the absurd. I can't think of any good reason that you couldn't do this with infinite dimensional matrices, and, furthermore, why you can't just do this forever. Like, an infinite dimensional matrix whose elements are infinite dimensional matrices whose elements are infinite dimensional matrices ad infinitum. I have a hard time conceptualizing how this would *be* anything sensible, but I thought of a context they might come up in. This idea is brand new to me, but wouldn't the Hamiltonian of a free particle in an infinite dimensional space be able to written in such a strange form? Seemingly even more bizarrely, the cardinality of the dimensions of each matrix would also then be uncountable. So, is there any sense to this nonsense? Or is this notion of some kind of horrid infinite matrix from hell complete rubbish?