#R initialize motrix mod#

When a concept comes out of mathematics it feels more authoritative, a deep fact about the logical structure of the universe, perhaps closer to the root of all the mysteries. Philosophically, P₇ is interesting because numbers-these existential things that seem to be around whether we think about them or not-have naturally formed into this “circular” shape. The cycle-of-7, when combined with other simple groups (also in matrix format), might model a biological system like a metabolic pathway. Three different number systems but they’re all essentially the same thing, which is this idea of a “cycle-of-7”. (Counting = money, or demography, or forestry matrix = classical mechanics, or video game visuals imaginary numbers = electrical engineering, or quantum mechanics.) are all demonstrating the same underlying logic.Īlthough each is merely an idea with only a spiritual existence, these are the kinds of “logical atoms” that build up the theories we use to describe the actual world scientifically. (A different one, but “the unit element” nonetheless.) Whoa! All of a sudden at the 7th step we’re back to “ 1” again. A third way of presenting the cyclic 7-group, which we can also do in R: > w w

#R initialize motrix mod#

Another way of presenting the group is with the pair, + mod 7 (that’s where it gets the name Z₇, because ℤ=the integers. The pair M.7, %*% is one way of presenting the only consistent multiplication table for 7 things. What you’ve just discovered is the cyclic group P₇ (also sometimes called Z₇). It would be exponent rules thing^x × thing^y = thing^ modulo 7.

Likewise if you multiplied intermediate matrices from midway through, you would still travel around within the cycle. If you multiplied again you would go through the cycle again. Look at the last one! It’s the identity matrix! Back to square one! Here are some more powers of M.7: > M.7 %^% 4 %*% M 131 times, I would need to use the expm package and then the %^% operator for the power. If I wanted to do straight-up matrix powers rather than typing M %*% M %*% M %*% M %*%. Look what happens when you multiply M.7 by itself: it starts to cascade. ( * does entry-by-entry multiplication, which is good for convolution but not for this.) Matrix multiplication in R is the %*% symbol, not the * symbol. Let’s call this matrix M.7 (a valid name in R) and look at the multiples of it. So the concatenated c(2,3,4,5,6,7,1) become the new row numbers. Now how about a connection to group theory?įirst take a 7-dimensional identity matrix, then rotate one of the rows off the top to the bottom row. Tikhonov regularisation is also a way of puffing air on a singular matrix ( det|M|=0) so as to make the matrix invertible without altering the eigenvalues too much.

That’s a form of penalty to rule out overly complex statistical models. īut while I have your attention, let’s do a couple mathematically interesting things with identity matrices.įirst of all you may have heard of Tikhonov regularisation, or ridge regression. You make identity matrices with the keyword diag, and the number of dimensions in parentheses. Hopefully this saves someone ten minutes of digging about in the documentation. I googled for this once upon a time and nothing came up.