Homework Statement
A long wire carries a current I and is centered in a long hollow cylinder of inner radius a and outer radius b. The cylinder is made of a linear material with permeability \mu. Find \mathbf{B} and \mathbf{H} everywhere.
Homework Equations
The Attempt at a...
I think you have the right idea for showing that 0 is not an eigenvalue of A, though you're not using the same names for the matrices as I used. The idea is that an invertible matrix remains invertible when you rotate it to a different basis because of the property of determinants that you...
I tried to edit the Latex, but it doesn't seem to have changed. I should have written, 'There's some rotation U s.t. U\ A'\ U^T\ is diagonal. Set B\ =\ U\ A'\ U^T\ .'
You're right - we can't assume that the eigenvalues of A are {x, -x}. I thought it might be possible to prove this, but I realize now that it's not true in general.
We do know, however, that 0 is not an eigenvalue of A (by non-degeneracy). So the problem is reduced to the following...
I don't know for sure that this works, but here's an idea. We know that A can be diagonalized. Suppose that the set of eigenvalues of A is {x, -x} for some positive real number x. Then the P we need is just some orthogonal matrix (ie, an improper rotation, det P = +1 or -1) multiplied by a...
I'm happy to help out if you correct a few errors first.
In (1): you have written 'A = 1.' But in fact your final answer has A = 0. Are you using 'A' to mean different things?
In (2): we know that y(0) = 1, so it can't be the case that 1 + A = 0.
Finally, your answer y(x) = 1 + sin (x) does...
The modulus of a complex number z is defined to be the positive square root of z z'. That's it - it's just a definition.
The motivation for this definition is that the modulus is supposed to be the length of z if you think of z as a vector in the x-y plane. In the case of z = 4 + 3i, the...