Abstract
There has recently been an intense discussion about the (re)discovery of a formula that relates the components of an orthonormal basis of eigenvectors of an Hermitian matrix A of order n to the eigenvalues of A and the eigenvalues of some sub-matrices of A of order \(n-1\). In this paper we will show that the above-mentioned formula is a consequence of the simultaneous diagonalization of a square matrix (not necessarily Hermitian) and its adjugate matrix.
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Reference
Denton, P.B., Parke, S.J., Tao, T., Zhang, X.: Eigenvectors from eigenvalues: a survey of a basic identity in linear algebra. arXiv:1908.03795v2[math.RA]
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Bertapelle, A., Candilera, M. Eigenvectors and eigenvalues: a new formula?. Boll Unione Mat Ital 13, 329–333 (2020). https://doi.org/10.1007/s40574-020-00222-z
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DOI: https://doi.org/10.1007/s40574-020-00222-z