Recent content by DavidK

Matt you don't happen to have a good reference for what you wrote above? You see, I'm using the fact that Tr(U^k) can't be zero for all k if U is unitary in a text I'm writing, but I do not want to litter the text with details regarding this fact. Any help would be highly appreciated.

I could mention that I have a very strong hunch that the largest possible value for n is N, and that this value is reached for unitaries of the form [U]_{kl} = \delta_{k+1 mod(N),l} .

Assume U is a NxN unitary matrix. Further assume that for all k<n: Tr(U^k)=0. What is the larges possible value for n?

I'm not 100% sure about the subtleties of the rules regarding when using your hands is regarded as unitentional. However, Nesta shoved the ball away from himself with his arm and hand. It was probably just a reflex movement, but the situation differs somewhat from ordinary situations judged as...

It is difficult to know if it was a conscious choice or not, it wasn't however obviously unintentional. I would have to see the situation again to be sure.

This is a crazy match. I don't think the referee sucks, however he has to make a lot of difficult decisions. He missed an obvious penalty kick though (Nesta used his hand). Regarding the US team, the swedish television commentators call their efforts heroic.

I think I have solved the problem for the 2\times2 case. A positive matrix A can in this case be expressed as: A=\frac{\mbox{Tr}(A)}{2}(I+r_x \sigma_x+r_y\sigma_y + r_z \sigma_z), where \sigma_x, \sigma_y, \sigma_z are the standard Pauli matrices, and \bar{r}_a=(r_x,r_y,r_z) is a...

An Hermitian matrix H is positive definite if all its eigenvalues are nonzero and positive. Assume that the matrices A,B are positve definite, and that the difference A-B is positve definite. Now, for which unitary matrices, U , is it true that the matrix A-UBU^{\dagger} is positve...

Ahhh...now it all makes sense again :approve: .

Thanks for the very informative answer. A natural follow up question is: why is the Earth at rest relative the CMB? Is it something one should expect?

Consider two farmes of reference moving relative each other. In one of the frames the CMD is fully isotropic, i.e., it looks the same in all directions. In the other frame however, the CMD should be red shifted in one direction and blue shifted in the other direction. Thus, the first frame can...

The fidelity can be viewed as a meassurement of distance between quantum states. If the fidelity is close to one, the states are close.

Yes I did :blushing:

In the finite dimensional case a linear operator, A, is diagonalizable if it is normal, i.e. AA^{\dagger}=A^{\dagger}A.

I would suggest that you use the formula: P(A|B)= \frac{P(B)P(B|A)}{P(A)} instead. In the first example A is "the box contains 10 balls" and B is "You pick ball number 9", we get P(B)= \frac{1}{2}(\frac{1}{10}+\frac{1}{100}), P(B|A)= \frac{1}{10}, and P(A)=\frac{1}{2}. Hence P(A|B)=...