The eigenvector-eigenvalue identity, which gives squared eigenvector components from matrix and minor eigenvalues, is a classical result with a long history of rediscovery, now compiled with proofs.
Billey and Bridget E
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Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra
The eigenvector-eigenvalue identity, which gives squared eigenvector components from matrix and minor eigenvalues, is a classical result with a long history of rediscovery, now compiled with proofs.