[Last edition: 28/07/2006]
As I promised two weeks ago, I'll talk about cubes today. One of the things Prof. Caticha taught me in his visit is an interesting way of representing cubes in any dimension using matrices. First, let me explain what is a n-dimensional cube, or simply a n-cube. Let us use an iterative method to construct it. Let us set up our notation: the representation of the n-cube will be denoted by the letter
The 0-dimensional cube is just a point and we will represent it by a 1x1 matrix, or as it is more commonly known, by a number: the number 1. We will consider n-cubes as being non-directed graphs and represent them by their connectivity (or incidence) matrix: the (i,j) element of the matrix will be 1 if the i-th vertex of the graph is connected with the j-th vertex and 0 otherwise. This justifies using the number 1 to represent the 0-cube, once it has just 1 point which we will consider as connected to itself.
Now, let us proceed two the 1-cube. The one-dimensional cube is just a line segment joining two points. We will see it as two 0-cubes joined by their vertices and then it will be represented by the 2x2 matrix:
In the end, a n-cube can be described by a block matrix derived from the (n-1)-cube with the form
Where I_{n-1} (in LaTeX notation :) ) is the (n-1) x (n-1) identity matrix and its role is to connect the vertices of the two C_{n-1} cubes to form its edges. An algebraic way to write this is
Where, if A is a 2x2 matrix, then
is the exterior, tensor or Kroenecker product between the two matrices A and B, which gives as a result a block matrix as indicated, and
is a Pauli matrix. Actually, I think there is a slightly more beautiful formula, but I forgot now and I'm waiting for Prof. Caticha to answer me an email I sent him...
Okay, now that I did what I promised, some more random things.
Latest (and some old) interesting news:
As I promised two weeks ago, I'll talk about cubes today. One of the things Prof. Caticha taught me in his visit is an interesting way of representing cubes in any dimension using matrices. First, let me explain what is a n-dimensional cube, or simply a n-cube. Let us use an iterative method to construct it. Let us set up our notation: the representation of the n-cube will be denoted by the letter
The 0-dimensional cube is just a point and we will represent it by a 1x1 matrix, or as it is more commonly known, by a number: the number 1. We will consider n-cubes as being non-directed graphs and represent them by their connectivity (or incidence) matrix: the (i,j) element of the matrix will be 1 if the i-th vertex of the graph is connected with the j-th vertex and 0 otherwise. This justifies using the number 1 to represent the 0-cube, once it has just 1 point which we will consider as connected to itself.
Now, let us proceed two the 1-cube. The one-dimensional cube is just a line segment joining two points. We will see it as two 0-cubes joined by their vertices and then it will be represented by the 2x2 matrix:
In the end, a n-cube can be described by a block matrix derived from the (n-1)-cube with the form
Where I_{n-1} (in LaTeX notation :) ) is the (n-1) x (n-1) identity matrix and its role is to connect the vertices of the two C_{n-1} cubes to form its edges. An algebraic way to write this is
Where, if A is a 2x2 matrix, then
is the exterior, tensor or Kroenecker product between the two matrices A and B, which gives as a result a block matrix as indicated, and
is a Pauli matrix. Actually, I think there is a slightly more beautiful formula, but I forgot now and I'm waiting for Prof. Caticha to answer me an email I sent him...
Okay, now that I did what I promised, some more random things.
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2 comments:
Hey, I kind of lost you where you hit the algebric equation with the circle with the x in it. What is that called and what does it mean?
Hi, Mac.
The circle with the x in the middle is the tensor product between the two matrices. You can find it in the Wikipedia by the link I put next to the equation. Anyway, I added a little explanantion in the post and corrected the algebraic equation that was inverted...
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