I googled for this once upon a time and nothing came up. Hopefully this saves someone ten minutes of digging about ter the documentation.
You make identity matrices with the keyword diag , and the number of dimensions te parentheses.
But while I have your attention, let’s do a duo mathematically interesting things with identity matrices.
Very first of all you may have heard of Tikhonov regularisation, or ridge regression. That’s a form of penalty to rule out overly complicated statistical models. @benoithamelin explains on @johndcook’s blog that
- Tikhonov regularisation is also a way of puffing air on a singular matrix ( det|M|=0 ) so spil to make the matrix invertible without altering the eigenvaluestoo much.
Now how about a connection to group theory?
Very first take a 7-dimensional identity matrix, then rotate one of the rows off the top to the bottom row.
Inwards the brackets it’s [row,katern]. So the concatenated c(Two,Three,Four,Five,6,7,1) become the fresh row numbers.
Let’s call this matrix M.7 (a valid name te R ) and look at the multiples of it. Matrix multiplication te R is the %*% symbol, not the * symbol. ( * does entry-by-entry multiplication, which is good for convolution but not for this.)
Look what happens when you multiply M.7 by itself: it starts to cascade.
If I desired to do straight-up matrix powers rather than typing M %*% M %*% M %*% M %*% . %*% M 131 times, I would need to use the expm package and then the %^% technicus for the power.
Here are some more powers of M.7 :
Look at the last one! It’s the identity matrix! Back to square one!
Or should I say square zero. If you multiplied again you would go through the cycle again. Likewise if you multiplied intermediate matrices from midway through, you would still travel around within the cycle. It would be exponent rules thing^x thing^y = thing^[x+y] modulo 7.
What you’ve just discovered is the cyclic group P (also sometimes called Z ). The pair M.7, %*% is one way of presenting the only consistent multiplication table for 7 things. Another way of presenting the group is with the pair <0,1,Two,Trio,Four,Five,6>, + mod 7 (that’s where it gets the name Z , because =the integers. A third way of presenting the cyclic 7-group, which wij can also do ter R :
Whoa! All of a unexpected at the 7th step we’re back to “ 1 ” again. (A different one, but “the unit element” nonetheless.)
- counting numbers,
- matrix-blocks, and
- a stadionring of imaginary numbers
— are all demonstrating the same underlying logic.
Albeit each is merely an idea with only a spiritual existence, thesis are the kinds of “logical atoms” that build up the theories wij use to describe the actual world scientifically. (Counting = money, or demography, or forestry, matrix = classical mechanics, or movie spel visuals, imaginary numbers = electrical engineering, or quantum mechanics.)
Three different number systems but they’re all essentially the same thing, which is this idea of a “cycle-of-7”. The cycle-of-7, when combined with other ordinary groups (also te matrix format), might proefje a biological system like a metabolic pathway.
Philosophically, P is interesting because numbers—these existential things that seem to be around whether wij think about them or not—have naturally formed into this “circular” form. 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.
Ter the real world I’d expect various other processes to hook into P —like a noise matrix, or some other groups. Other fundamental units should combine with it, I’d expect to see P instantiated by itself infrequently.
Mathematically, P is interesting because three totally different number systems (imaginary, counting, square-matrix) are shown to have one “root cause” which is the group concept.
John Rhodes got famous for arguing that everything, but EVERYTHING, is built up from a logical structure made from SNAGs, of which P =C =Z is one. viz, algebraic engineering
Ter brief, groups are one of those things that make people think: Hey, man, maybe EVERYTHING is a matrix. I’m going to go meditate on that.