Silly question about matrices with matrix elements

In summary, the conversation discusses the concept of a matrix whose elements are also matrices, and the possibility of extending this idea to infinite dimensional matrices. It is suggested that this idea may have applications in Quantum Mechanics, particularly in relation to Landauer's principle. However, it is also acknowledged that this concept may not have any practical use.
  • #1
Monocles
466
1
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
  • #2
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.
 
  • #3
Aww, that's a little more boring than I had hoped. Oh well!
 
  • #4
I am sorry, I can't help it :P
 
  • #5
@Landau,
Hey, I hope u know how to erase it :D
 
  • #6
I'm afraid I don't understand you.
 
  • #7
NaturePaper said:
@Landau,
Hey, I hope u know how to erase it :D
:confused:
 
  • #8
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" [Broken]. 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:

1. What is a matrix?

A matrix is a rectangular array of numbers or variables arranged in rows and columns. It is used to represent data or perform mathematical operations.

2. What are matrix elements?

Matrix elements are the individual numbers or variables within a matrix. They are usually represented by a subscript indicating their position in the matrix (e.g. aij is the element in the ith row and jth column).

3. Can a matrix have different types of elements?

Yes, a matrix can have elements that are numbers, variables, or even other matrices. However, for mathematical operations to be performed on a matrix, all of its elements must be of the same type.

4. What is the purpose of matrices in science?

Matrices are used in various scientific fields, including physics, chemistry, and computer science. They are particularly useful for representing and manipulating large datasets, solving systems of equations, and performing transformations in geometric and quantum systems.

5. Are there any special properties of matrix elements?

Yes, matrix elements can have certain properties depending on their position in the matrix. For example, the diagonal elements (aii) in a square matrix are known as eigenvalues and have important applications in linear algebra and quantum mechanics.

Similar threads

  • Linear and Abstract Algebra
Replies
17
Views
4K
  • Linear and Abstract Algebra
Replies
6
Views
1K
Replies
4
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
886
Replies
2
Views
4K
  • Science and Math Textbooks
Replies
27
Views
2K
  • Linear and Abstract Algebra
Replies
4
Views
5K
  • Set Theory, Logic, Probability, Statistics
Replies
14
Views
1K
  • MATLAB, Maple, Mathematica, LaTeX
Replies
13
Views
2K
  • Linear and Abstract Algebra
Replies
2
Views
4K
Back
Top