Silly question about matrices with matrix elements

[Monocles]

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?

 

[Landau]

A 2x2 matrix whose elements are 4x4 matrices is nothing special. It is just a 8x8 matrix, where the four 4x4 'blocks' are considered seperately for some reason. See e.g. "Block Matrix" at Wolfram.

In the same way, any matrix is a 1x1 matrix with as one (only) entry the matrix itself.

 

[Monocles]

Aww, that's a little more boring than I had hoped. Oh well!

 

[Landau]

I am sorry, I can't help it :P

 

[NaturePaper]

@Landau,
Hey, I hope u know how to erase it :D

 

[Landau]

I'm afraid I don't understand you.

 

[Mark44]

@Landau,
Hey, I hope u know how to erase it :D

 

[NaturePaper]

Well, I thought (and still think) Landau is a specialist in Quantum Mechanics (which is sometimes called the mathematical modelling of nature and uses lots of linear operators:P). Particularly, the whole matrix theory is a basic tool in finite dimensional QM and its modern branch Quantum Information Theory. In this new branch there is a basic principle, http://en.wikipedia.org/wiki/Landauer%27s_principle" . As the name is a bit different from Landau, I have written the above line for fun. I am sorry if it somehow causes inconvenience to anyone.

Cheers,
NP