At the other extreme, if is equal to a different eigenvalue of, then now appears as an eigenvalue of, and both sides of (1) now vanish. If it is equal to, then the eigenvalues of are the other eigenvalues of, and now the left and right-hand sides of (1) are equal to. The upper left entry of is one of the eigenvalues of. Suppose for instance that is a diagonal matrix with all distinct entries. One can get some feeling of the identity (1) by considering some special cases. I also give two other proofs below the fold, one from a more geometric perspective and one proceeding via Cramer’s rule.) It was certainly something of a surprise to me that there is no explicit appearance of the components of in the formula (1) (though they do indirectly appear through their effect on the eigenvalues for instance from taking traces one sees that ). (I do so below the fold we ended up not putting this proof in the note as it was longer than the two other proofs we found. In the random matrix theory literature, for instance in this paper of Erdos, Schlein, and Yau, or this later paper of Van Vu and myself, a related identity has been used, namelyīut it is not immediately obvious that one can derive the former identity from the latter. But perhaps it is surprising that such a formula exists at all one does not normally expect to learn much information about eigenvectors purely from knowledge of eigenvalues. Once one is aware of the identity, it is not so difficult to prove it we give two proofs, each about half a page long, one of which is based on a variant of the Cauchy-Binet formula, and the other based on properties of the adjugate matrix. įor some real number, -dimensional vector, and Hermitian matrix, then we haveĪssuming that the denominator is non-zero.
Where is the Hermitian matrix formed by deleting the row and column from. Let be a unit eigenvector corresponding to the eigenvalue, and let be the component of. Theorem 1 Let be an Hermitian matrix, with eigenvalues. This note gives two proofs of a general eigenvector identity observed recently by Denton, Parke and Zhang in the course of some quantum mechanical calculations. Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv the short unpublished note “ Eigenvectors from eigenvalues“.
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