Thread: 1 = -1?

Originally Posted by Xei

I don't know about this, want to talk about it some more?

Sure. Like I said I'm rusty and this is all very informal, but think about it like this. Take the most basic linear equation in one dimension w = cz. We are here interpreting number in two ways, as a function and as an element in the domain/range. Of the two numbers c is playing the role of a function and z is playing the role of an element.

So when we go to generalize to higher dimensions (and solve more equations) we have two distinct roles to fulfill. One gets fulfilled with vectors and the other with matrices. So we end up with something like w=Cz.

The analogy is however still striking. Take matrices. if M is a matrix ring over R then we can define maps M -> M. Take the map A |-> A². Differentiate it. The derivative at A will be a linear function on H in M where H is interpreted as the displacement. The derivative is H |-> AH + HA! Consider that if A and H commute than this is just 2AH. But in one dimension, x |-> x² has a derivative at x which is a linear function on h real of the form h |-> 2xh.

So the analogy can be very tight at times. It's of course important to realize that matrices are different from numbers in that everything doesn't carry over by a long shot but there is some intuition to be had there.

As far as grassman algebras and differential forms go, they're ways to generalize the purpose of numbers in measuring various volumes. They're pretty deep for me but I could probably bullshit my way through an explanation.

Also out of curiosity where and when do you get all of this learning?

Books. Ending mostly about seven years ago or so. Though I do try to at least keep sharp with my basic algebra and analysis. I wasn't above dropping in on a math department and asking random people if I got stuck with something. I'm actually very slow and stupid so it's not like I'm a genius or anything. At least not at math Nobody was impressed with anything other than that I work very hard.

Another example is positive and negative matrices. For the reals, the invertible elements are in an open set with two disconnected components, N = (-inf, 0) and P = (0, inf). Numbers in P multiplied by each other end up in P so it's closed under multiplication. Numbers in N multiplied together end up in P. Multiplying a number in P and a number in N ends up in N. This is just the negative times a negative is a positive rule and its variants.

Now consider the invertible elements of some matrix ring over the reals. These will be the matrices with non-zero determinants. Now N is the set of matrices with Det(A) < 0 and P will have Det(A) > 0. Note that Det(AB) = Det(A)Det(B) so matrix multiplications satisfies all the same negative-times-a-negative-is-a-positive type rules.

Also note that it is an open set with two disconnected components. To see that it's open, just note that the determinant is a polynomial. Because polynomials are continuous, given an invertible matrix A, we can put a sufficiently small open ball around it so that the determinate is non-zero on that too.

To show that it's disconnected, note that a continuous function that switches signs must take on a value of zero. Hence any path s |-> A(s) that we can draw from a positive matrix to a negative matrix or vice-versa will have Det(A(s)) = 0 for some s. Here the s are just real numbers.

So again, it's very much like numbers.

no wai.