Silly question about matrices with matrix elements

Click For Summary

Discussion Overview

The discussion centers around the concept of matrices whose elements are themselves matrices, particularly exploring the implications of infinite dimensional matrices in quantum mechanics. Participants consider both the theoretical aspects and potential applications of such constructs.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant introduces the idea of a 2x2 matrix with 4x4 matrix elements and speculates about extending this to infinite dimensional matrices, questioning the sensibility of such constructs.
  • Another participant argues that a 2x2 matrix with 4x4 elements is essentially an 8x8 matrix, suggesting that the notion is not particularly special and refers to it as a "Block Matrix."
  • A participant expresses disappointment that the concept is less novel than anticipated, indicating a desire for more complexity.
  • There are several off-topic exchanges that do not contribute to the main discussion about matrices.
  • One participant references Quantum Mechanics and Quantum Information Theory, noting the relevance of matrix theory in these fields, while also making a light-hearted comment about a name similarity.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the significance or utility of infinite dimensional matrices, with some viewing it as a trivial extension and others expressing curiosity about its implications.

Contextual Notes

The discussion includes informal exchanges and references that may not directly relate to the mathematical concepts being explored. There is also a lack of formal definitions or clarifications regarding the properties of infinite dimensional matrices.

Who May Find This Useful

Readers interested in quantum mechanics, linear algebra, and the theoretical implications of matrix theory may find this discussion relevant.

Monocles
Messages
463
Reaction score
2
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?
 
Physics news on Phys.org
A 2x2 matrix whose elements are 4x4 matrices is nothing special. It is just a 8x8 matrix, where the four 4x4 'blocks' are considered separately 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.
 
Aww, that's a little more boring than I had hoped. Oh well!
 
I am sorry, I can't help it :P
 
@Landau,
Hey, I hope u know how to erase it :D
 
I'm afraid I don't understand you.
 
NaturePaper said:
@Landau,
Hey, I hope u know how to erase it :D
:confused:
 
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
 
Last edited by a moderator:

Similar threads

Graduate Structure of modulo-integer matrix group GL(r,Z(n))?
  • · Replies 17 ·
Replies
17
Views
7K
Graduate How to determine an infinitely dimensional matrix is positive definite
  • · Replies 2 ·
Replies
2
Views
4K
Graduate How Do You Set Up Matrices for Cubic Spline Interpolation?
  • · Replies 13 ·
Replies
13
Views
5K
Undergrad A question I have about my L.A. textbook (2)
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Question about a matrix of vector element ratios
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Question about commutator involving fermions and Pauli matrices
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Help: All subspaces of 2x2 diagonal matrices
  • · Replies 4 ·
Replies
4
Views
6K
Graduate Diagonalization of 2x2 Hermitian matrices using Wigner D-Matrix
  • · Replies 4 ·
Replies
4
Views
5K
Maximum determinant of matrix with only 1 and -1 elements?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Diagonalization of 8x8 matrix with Euler angles
  • · Replies 2 ·
Replies
2
Views
5K