I'm trying to understand mathematical intuition of HHL algorithm using original paper For now I stuck at the part of Phase estimation. If I understand correctly, if vector b is the eigenvector of the operator, then the phase estimation produces phase of a certain eigenvalue of this operator at the second register and does not change the first register (with vector b) due to the definition of eigenvalues and eigenvectors. But what happens in the HHL algorithm? Why QPE does still produces the phase eigenvalues even if b is not and eigenvector anymore? Does the vector b itself changes or what do the words

decompose |b> in the eigenvector basis

mean? Additionally, what does the equation below mean, how was it created and where the variables come from - this is not clear for me also.

I'm trying to understand mathematical intuition of HHL algorithm using original paper (arXiv) For now I stuck at the part of Phase estimation. If I understand correctly, if vector b is the eigenvector of the operator, then the phase estimation produces phase of a certain eigenvalue of this operator at the second register and does not change the first register (with vector b) due to the definition of eigenvalues and eigenvectors. But what happens in the HHL algorithm? Why QPE does still produces the phase eigenvalues even if b is not and eigenvector anymore? Does the vector b itself changes or what do the words

decompose $|b\rangle$ in the eigenvector basis

mean? Additionally, what does the equation below mean, how was it created and where the variables come from - this is not clear for me also. $$ |\Psi_0\rangle:=\sqrt{\frac2T}\sum_{\tau=0}^{T-1}\sin\frac{\pi(\tau+\frac12)}T|\tau\rangle $$