Eigenvectors and eigenvalues: a new formula?

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.