By writing A in terms of the real and imaginary parts is really more illuminating. It is simply too bothersome not to be able to make use of this kind of symmetry on a problem of such general interest. As you mentioned, maybe there's a way to restart the iteration when failure happens, however...
You are totally right. That's what I worry about. I was just hoping maybe the solution is still valid given the particular problem I have (with further constraints on the system). Remember the solution is an iterative one. My previous post has a lot more details. Can you help look at it? Thank...
Hi chiro, thanks for answering. Le me first clarify a few points:
1. The Gram–Schmidt process I mentioned is actually Anordi Gram–Schmidt process, where you start with an (arbitrary) initial vector v, and try to find an orthogonal basis for the Krylov space K=span{v,Av,A^2v,...}.
2. The...
It is well known that a Hermitian symmetric complex matrix A, A^{\dagger}=A can be taking into a tridiagonolized form: A=V^{\dagger}HV where ^{\dagger} is Hermitian conjugate and H is the tridiagonal Hessenberg matrix, and V^{\dagger}V=VV^{\dagger}=I. This decomposition is realized using Schmidt...
Homework Statement
Let f: X->R be measurable, prove that Z={(x,y)|y=f(x)} is a measurable set of XxR.
Homework Equations
A subset Z of XxR is measurable iff Z is a countable union of product of measurable sets of X and R.
The Attempt at a Solution
Let R=\cup_kV_k, where V_k are...
This is the book I use
http://books.google.com/books/about/Some_modern_mathematics_for_physicists_a.html?id=9PXuAAAAMAAJ
The definition of general measurable space in this book
Definition 7.1(3). Let X be a (universal) set and let psi be a sigma-ring on X which has the property that X is a...
But A^c need not be measurable in a general measurable space, which is not necessarily a Borel field, only a \sigma-ring whose union is X. Am I completely wrong?
Homework Statement
Prove that the characteristic function \chi_A: X\rightarrow R, \chi_A(x)=1,x\in A; \chi_A(x)=0, x\notin A, where A is a measurable set of the measurable space (X,\psi) , is measurable.
Homework Equations
a function f: X->R is measurable if for any usual measurable set...
Since I just started reading SR, I only want to make a comment. Since the lightspeed train is accelerating by going around the Earth which is assumed to be an inertial frame, the train is certainly not an inertial frame, therefore the time dilation should be modified in such a system. Is it...
Griffiths also mentioned another paradox earlier in his book regarding a point charge's radiation reaction, where he derived a dumbbell shaped accelerating charge having infinite mass, and he mentioned in the footnote that such paradox is "covered up" under quantum EM. His derivation of that...
Thank you. Are Maxwell's equations as well as Lorentz force law completely compatible, assuming there's no "point charge", only continuous distribution of charges exists?
Thank you, that answers my first question and leads to the second one, i.e., if I define a inertial reference frame in which Maxwell equations as well as SR hold and ignore gravity. Is this "mathematical" framework completely self-consistent? Do we need to modify Newton's second and third law?
Thank you for replying, since I don't know Lorentz transformation (or anything about relativity), let me elaborate what I have in mind:
Assuming I'm a person standing at the conducting surface measuring EM field fluctuations, if the source of the plane wave is moving toward me, I would observe...
I have no background in relativity. Recently I started reading some introduction to special relativity in Griffith's EM book, where he vaguely defined an inertial reference frame as one in which Newton's first law holds. Now according to this definition, does such frame exist in nature?
On...
A plane wave normally incident onto a perfectly conductive surface moving in the normal direction with constant velocity comparable to the speed of light. How do I solve such problem? If I treat the conductor as static, and the source of plane wave as a moving source, do I only need to consider...