Recent content by xuphys

Hi, When I do the following transformation: $$ X_1=x_1+x_2 \\ X_2=x_2 $$ It turns out that the Jacobian ##\partial (X_1,X_2)/\partial (x_1,x_2)## is 1. But we have: $$ dx_1dx_1+dx_1dx_2=d(x_1+x_2)dx_2=dX_1dX_2=|\partial (X_1,X_2)/\partial (x_1,x_2)|dx_1dx_2=dx_1dx_2 $$ So we...

OK. What I thought is, since A and B are commutative, and they are also hermitians, so AB is also a hermitian (easy to prove). Then AB is diagonalizable: AB=UDU-1 i.e. D=U-1ABU=(U-1AU)(U-1BU) Now I am unable to prove that both U-1AU and U-1BU are diagonal... and I am not sure...

Thanks for your reply. What I would say is that this is a common mathematical theorem which is one of the mathematical basis in quantum mechanics, but not a textbook style problem. I ask here because my textbook (which is not written in English) only gives a simplified version of proof...

Hi, Does anyone know how to prove that two commutative Hermitian matrices can always have the same set of eigenvectors? i.e. AB - BA=0 A and B are both Hermitian matrices, how to prove A and B have the same set of eigenvectors? Thanks!

Thank you for responding. Do you know about dynamical apparent horizon? It seems related to cosmological event horizon. I am wondering whether the dynamical apparent horizon is the same thing with what you mentioned here... Thank you again!

What's the difference between apparent horizon and event horizon? I checked Wikipedia but I still don't understand. Could anyone give a short explanation? Thanks!

It takes a lot of time for my Fortran program to do I/O jobs. Is there any way to do I/O jobs while at the same time it is doing computing jobs? To do I/O jobs and computing jobs at the same time would save a lot of time. Thanks.