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I have some perplexity concerning the concept of phase estimation: by definition, given a unitary operator U$U$ and an eigenvector |u>$|u\rangle$ with related eigenvalue exp$\text{exp}(2πiφ2\pi i \phi)$, the phase estimation allows to find the value of φ$\phi$. This would mean that I would be able to determine an eigenvalue of a certain matrix given that I know already one of its eigenvectors? But isn't the fact that needing an eigenvector beforehand would quite reduce the usefulness of the phase estimation itself?

I have some perplexity concerning the concept of phase estimation: by definition, given a unitary operator U and an eigenvector |u> with related eigenvalue exp(2πiφ), the phase estimation allows to find the value of φ. This would mean that I would be able to determine an eigenvalue of a certain matrix given that I know already one of its eigenvectors? But isn't the fact that needing an eigenvector beforehand would quite reduce the usefulness of the phase estimation itself?

I have some perplexity concerning the concept of phase estimation: by definition, given a unitary operator $U$ and an eigenvector $|u\rangle$ with related eigenvalue $\text{exp}(2\pi i \phi)$, the phase estimation allows to find the value of $\phi$. This would mean that I would be able to determine an eigenvalue of a certain matrix given that I know already one of its eigenvectors? But isn't the fact that needing an eigenvector beforehand would quite reduce the usefulness of the phase estimation itself?
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What is the actual power of Quantum Phase Estimation?

I have some perplexity concerning the concept of phase estimation: by definition, given a unitary operator U and an eigenvector |u> with related eigenvalue exp(2πiφ), the phase estimation allows to find the value of φ. This would mean that I would be able to determine an eigenvalue of a certain matrix given that I know already one of its eigenvectors? But isn't the fact that needing an eigenvector beforehand would quite reduce the usefulness of the phase estimation itself?