Normal Matrices Examples (URGENT)
I need to produce 2 x 2 normal matrices A and B such that A + B is not normal. I have proven that AB is normal if AB = BA using the Householder matrix form. But I can't find a form for A + B failing to be normal.
A matrix A with entries a, b, c, d is going...
If each σ is diagonal or anti-diagonal, could I just write each diagonal entry of exp(σ) as e^σij, then add all of the σ's up? Like this case:
http://en.wikipedia.org/wiki/Matrix_exponential#Diagonalizable_case
It's problem #5 on this homework set: https://docs.google.com/open?id=0B9c8sp75B5ZRMHAxYXB3MWdhYk0
I can calculate (\pi/4)(n1σ1 + n2σ2 + n3σ3) easily, but I have NO clue how a matrix M = exp[(\pi/4)(n1σ1 + n2σ2 + n3σ3)].
Well if I plug those z's in, I get z(-∞)/(z-∞) for m(z). I seem to hit a wall there. Same with n(z). I don't know how to fix those equations into a form that I can use for the composition.
Homework Statement
H is the upper-half plane model of the hyperbolic space
Find all Mobius transformations that send M to N.
Homework Equations
a) M = {0, 1, ∞}, N = {∞, 0, 1}
b) M = {0, 1, ∞}, N = {0, ∞, 2}
c) M = {i, -i, 3i}, N = {∞, i + 1, 6}
The Attempt at a Solution...
Homework Statement
Find all the fixed points to the following Mobius transformation.
Homework Equations
m(z) = (2z + 5)/(3z - 1)
The Attempt at a Solution
Aren't all fixed points going to map to themselves? So shouldn't it be solving for m(z) = z and coming up with roots of a quadratic...
You can absolutely prove that x is irrational and therefore xc is irrational and not an integer.
1. Show x is irrational by contradiction.
2. Assume a < xc < b for integers a, b, c, where xc is an integer.
3. Prove that since x is irrational, then xc is also irrational and therefore not an...