It actually turned out that the result was old, and had been rediscovered multiple times: https://terrytao.wordpress.com/2019/12/03/eigenvectors-from-eigenvalue

It actually turned out that the result was old, and had been rediscovered multiple times: https://terrytao.wordpress.com/2019/12/03/eigenvectors-from-eigenvalues-a-survey-of-a-basic-identity-in-linear-algebra/ https://terrytao.wordpress.com/2019/12/03/eigenvectors-from-...