Transformation of Box Constraints After PCA

It is a little bit hard to answer withouth further information : why would you like to impose box constraints after the PCA ? Did you impose them before the PCA or is it something you observed in the data ?Indeed, there is no guarantee that the output of the PCA (i.e. the coordinates of the points on the principal axis) will follow the initial box constraints.So you will have to rescale the output of your PCA to match the box constraints, just as you would do (did?) it before the PCA.

1. Observable: possible outcome of measurement vs (linear) transformation

I think that your problem is the question of collapse. Collapse of wavefunction is somewhat far from all the mathematical structure of quantum mechanics and was devised by Von Neumann as a further add-on to the formalism. The understanding of it in the framework of quantum mechanics formalism is something that, so far, has singled out a possible successful framework in the idea of decoherence. The idea of collapse can be seen as somewhat at the boundary when one understands that the following process$$|psi

angle=sum_nc_n|n

angle

ightarrow |k

angle$$lies somewhat far from a linear formalism.Much of the research work today in the area of the interpretation of quantum mechanics is rooted in a way or another into a clear framework to fit the bill for the collapse

2. Standard Normal Distribution Transformation Z=lnY

If $Z textN(0,1)$ variable with $Y=e^Z$. Then $ F_Y(y) = Pr(Y.

3. Is a Fourier transform a change of basis, or is it a linear transformation?

Rasmus' answer is probably what you want. But remember that the Fourier transform can also be defined in a discrete domain, and we have the DTFT and the DFT. The latter, finite, maps a sequence of N complex numbers to other complex sequence of same length, via a linear unitary transform. In this case (and, perhaps, only in this case) we can confidently say that it's a "change of basis".

4. Equivalence of Sobolev Norms via Fourier Transformation

Yes. Distinguish the cases $|xi|leq 1$ and $|xi|> 1$ and use crude bounds

5. Probability (Transformation Method)

To begin with: You probably wanted to write $F_X(x)= begincases 1-e^-lambda x & x>0 0 & x le 0 endcases$, otherwise it is not good distribution functionYou should not use the method to obtain density of $Y$ simply because $Y$ does not have density at all. Note that $Y$ only takes values from $mathbb N$ so to find its distribution it is enought to find values $mathbb P(Y=n)$ for any $n in mathbb N$. By definition, for $n in mathbb N$ we get: $Y=n$ if and only if $X in [n,n1)$. So:$mathbb P(Y=n) = mathbb P(X in [n,n1)) = mathbb P(X

6. Linear transformation that preserves the determinant

The conclusion you indicate is obtained as the main result in the following paper, but with an apparently stronger hypothesis: (EDIT: Not stronger at all, actually - just realized you are assuming the map is linear.)Determinant preserving maps on matrix algebrasGregor Dolinar and Peter SemrlLinear Algebra and its Applications Volume 348, Issues 1-3, 15 June 2002, Pages 189-192Let $M_n$ be the algebra of all $ntimes n$ complex matrices. If $phi:M_nM_n$ is a surjective mapping satisfying $det(Alambda B)=det(phi(A)lambdaphi(B))$ then either $phi$ is of the form $phi(A)=MAN$ or $phi$ is of the form $phi(A)=MA^TN$ where $M,N$ are nonsingular matrices with $det(MN)=1$.

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