There is a particular scenario that I am having trouble explaining to another person. I take my best attempt at explaining this scenario below.
Is there some mathematical concept / description that is more concise? I'm looking for a name, a list of potentially applicable names, or simply a less verbose treatment for this scenario.
Describing the Instance
Imagine that I have a set of data that can be graphically represented in a 2-D matrix. All of the blocks that aren't specified explicitly aren't occupied / are occupied by 0 (NULL) elements, such as shown below:
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| A | B | |
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| | C | D |
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| | E | |
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When I replicate this matrix three times to create a 2 x 2 matrix of the original 2-D matrix, I can represent the result as shown below:
--------------------------
| A | B | || A | B | |
--------------------------
| | C | D || | C | D |
--------------------------
| | E | || | E | |
--------------------------
--------------------------
| A | B | || A | B | |
--------------------------
| | C | D || | C | D |
--------------------------
| | E | || | E | |
--------------------------
I can specify a new set of data of the same dimensions within this set of replicated 2-D matrices. To do this, I simply specify a new matrix within the replicated 2-D matrices by drawing new boundaries (high-lighted in bold) in my graphical representation, as shown below:
-------------------------- | A | B | || A | B | | -------------------------- | | C | D || | C | D | -------------------------- | | E | || | E | | -------------------------- -------------------------- | A | B | || A | B | | -------------------------- | | C | D || | C | D | -------------------------- | | E | || | E | | --------------------------
It is important to note that in this new set of data, not only are the dimensions identical to the dimensions of the original set of data, but the contents are identical as well (there is 1 'A', 1 'B', 1 'C', etc.). The only difference is that the ordering of the contents has changed.
I've played around with this in larger 2-D sets of data and even some small 3-D sets of data, and I've always arrived at the same result:
- The contents are always identical (an
Ais never missing, there are never 2A's in the "new" set of data) - The order of those elements is almost always different (if the
Ais in the top corner of the "original" matrix, then it is not in the top corner of the "new" matrix)
I presume that because this is a rather simple concept at heart that there must be plenty of very satisfying mathematical ways of describing it already out in circulation.
I can't seem to put together the words to find those descriptions online. For some reason - perhaps the concept is not normally presented by means of this scenario - I cannot think of a mathematical term / concept to describe it. Neither could I come up with an intelligible Google search that didn't overflow beyond the length of the query text box. For the above reasons, I can't think of which tags are most appropriate for this question.
If $A,B, \dots$ are numbers, then this can be described with permutation matrices: Let $$J=\begin{pmatrix} 0&0 &1\\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix},$$ then multplication of a matrix $M$ (like the $3 \times 3$ in your question) with $J$ from the right shifts all columns one step to the right. I.e. the second column of $M$ becomes the first of $MJ$, the third of $M$ is the second of $MJ$ and so on. Likewise, multplication from the left shifts columns down, i.e. the last row of $M$ becomes the first of $JM$ and so on.
If I am not mistaken, the bolded matrix is the original matrix, just shifted one step to the right and one step down, i.e. it's $JMJ$.
If $A, B$ and so on are not numbers but square matrices themselves, one needs to build $J$ as a block matrix with identity matrices instead of the numbers $1$ above.
If we are interested in solutions for larger matrices than $3 \times 3$, then the key in construction $J$ is to place $1$s on the off-diagonal and a $1$ in the 'opposite' corner so that each row and column has a $1$.