|This is me right now. *snort*|
I've been studying for my PhD comprehensive exams recently. Today I've been reading about linear equations. Specifically, permutation matrixes. Matrixes are just columns and rows of numbers, like this:
This is a 4*4 matrix, but you can have matrices of all sorts of sizes, from 2* 2 to 1,587,47294 * 65 (for example. We can even visualize this matrix with colors, like this:
As you can see, 0 is dark blue, 1 is royal blue, and so on until you get to 10, which is dark red.
Now that we understand matrices, let's look at a permutation matrix. Permutation matrices are basically just matrices that have exactly one 1 in each row and column, and all other numbers are zero. Like this:
And here's the color representation of this matrix:
Again, zero is in dark blue, and 1 is royal blue.
The cool thing about permutation matrixes is that, when you multiply a regular matrix by a permutation matrix, it switches the rows of the regular matrix around. It's kind of like when you flip a hexaflexagon, you get a different color.
Now I'm going to multiply my original matrix by my permutation matrix:
Every time I multiply by my permutation matrix, the rows are moved around, until on the third multiplication, we are back at the original matrix!
What happens if I use a different permutation matrix, like this one:
This time, it takes four multiplications to get back to where we started:
What about this permutation matrix?
This time, no matter how many times I multiply the matrix, nothing happens!
What's going on here?
Well, the 1 in the permutation matrix tells us how the columns will be moved around. In the first example, for row one, the 1 was in column 4, so rows four and one switched. The second 1 was in column 1, so rows two and one switched. The third one was in column 3, so row three stayed where it was. Finally, the fourth 1 was in column 2, so rows four and two switched (remember, four had already switched with one, so row one is now in column 2. Here it is again:
|Permutation, why you no permutate?|
Here's a hexaflexagon pattern to try at home.