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Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv a completely rewritten version of our previous paper, now titled “Eigenvectors from Eigenvalues: a survey of a basic identity in linear algebra“. This paper is now a survey of the various literature surrounding the following basic identity in linear algebra, which we propose to call the eigenvector-eigenvalue identity:

Theorem 1 (Eigenvector-eigenvalue identity) Let ${A}$ be an ${n \times n}$ Hermitian matrix, with eigenvalues ${\lambda_1(A),\dots,\lambda_n(A)}$. Let ${v_i}$ be a unit eigenvector corresponding to the eigenvalue ${\lambda_i(A)}$, and let ${v_{i,j}}$ be the ${j^{th}}$ component of ${v_i}$. Then

$\displaystyle |v_{i,j}|^2 \prod_{k=1; k \neq i}^n (\lambda_i(A) - \lambda_k(A)) = \prod_{k=1}^{n-1} (\lambda_i(A) - \lambda_k(M_j))$

where ${M_j}$ is the ${n-1 \times n-1}$ Hermitian matrix formed by deleting the ${j^{th}}$ row and column from ${A}$.

When we posted the first version of this paper, we were unaware of previous appearances of this identity in the literature; a related identity had been used by Erdos-Schlein-Yau and by myself and Van Vu for applications to random matrix theory, but to our knowledge this specific identity appeared to be new. Even two months after our preprint first appeared on the arXiv in August, we had only learned of one other place in the literature where the identity showed up (by Forrester and Zhang, who also cite an earlier paper of Baryshnikov).

The situation changed rather dramatically with the publication of a popular science article in Quanta on this identity in November, which gave this result significantly more exposure. Within a few weeks we became informed (through private communication, online discussion, and exploration of the citation tree around the references we were alerted to) of over three dozen places where the identity, or some other closely related identity, had previously appeared in the literature, in such areas as numerical linear algebra, various aspects of graph theory (graph reconstruction, chemical graph theory, and walks on graphs), inverse eigenvalue problems, random matrix theory, and neutrino physics. As a consequence, we have decided to completely rewrite our article in order to collate this crowdsourced information, and survey the history of this identity, all the known proofs (we collect seven distinct ways to prove the identity (or generalisations thereof)), and all the applications of it that we are currently aware of. The citation graph of the literature that this ad hoc crowdsourcing effort produced is only very weakly connected, which we found surprising:

The earliest explicit appearance of the eigenvector-eigenvalue identity we are now aware of is in a 1966 paper of Thompson, although this paper is only cited (directly or indirectly) by a fraction of the known literature, and also there is a precursor identity of Löwner from 1934 that can be shown to imply the identity as a limiting case. At the end of the paper we speculate on some possible reasons why this identity only achieved a modest amount of recognition and dissemination prior to the November 2019 Quanta article.

Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv the short unpublished note “Eigenvectors from 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. The identity is as follows:

Theorem 1 Let ${A}$ be an ${n \times n}$ Hermitian matrix, with eigenvalues ${\lambda_1(A),\dots,\lambda_n(A)}$. Let ${v_i}$ be a unit eigenvector corresponding to the eigenvalue ${\lambda_i(A)}$, and let ${v_{i,j}}$ be the ${j^{th}}$ component of ${v_i}$. Then

$\displaystyle |v_{i,j}|^2 \prod_{k=1; k \neq i}^n (\lambda_i(A) - \lambda_k(A)) = \prod_{k=1}^{n-1} (\lambda_i(A) - \lambda_k(M_j))$

where ${M_j}$ is the ${n-1 \times n-1}$ Hermitian matrix formed by deleting the ${j^{th}}$ row and column from ${A}$.

For instance, if we have

$\displaystyle A = \begin{pmatrix} a & X^* \\ X & M \end{pmatrix}$

for some real number ${a}$, ${n-1}$-dimensional vector ${X}$, and ${n-1 \times n-1}$ Hermitian matrix ${M}$, then we have

$\displaystyle |v_{i,1}|^2 = \frac{\prod_{k=1}^{n-1} (\lambda_i(A) - \lambda_k(M))}{\prod_{k=1; k \neq i}^n (\lambda_i(A) - \lambda_k(A))} \ \ \ \ \ (1)$

assuming that the denominator is non-zero.

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. 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. 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, namely

$\displaystyle |v_{i,1}|^2 = \frac{1}{1 + \| (M-\lambda_i(A))^{-1} X \|^2}, \ \ \ \ \ (2)$

but it is not immediately obvious that one can derive the former identity from the latter. (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. 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 ${a,X}$ components of ${A}$ in the formula (1) (though they do indirectly appear through their effect on the eigenvalues ${\lambda_k(A)}$; for instance from taking traces one sees that ${a = \sum_{k=1}^n \lambda_k(A) - \sum_{k=1}^{n-1} \lambda_k(M)}$).

One can get some feeling of the identity (1) by considering some special cases. Suppose for instance that ${A}$ is a diagonal matrix with all distinct entries. The upper left entry ${a}$ of ${A}$ is one of the eigenvalues of ${A}$. If it is equal to ${\lambda_i(A)}$, then the eigenvalues of ${M}$ are the other ${n-1}$ eigenvalues of ${A}$, and now the left and right-hand sides of (1) are equal to ${1}$. At the other extreme, if ${a}$ is equal to a different eigenvalue of ${A}$, then ${\lambda_i(A)}$ now appears as an eigenvalue of ${M}$, and both sides of (1) now vanish. More generally, if we order the eigenvalues ${\lambda_1(A) \leq \dots \leq \lambda_n(A)}$ and ${\lambda_1(M) \leq \dots \leq \lambda_{n-1}(M)}$, then the Cauchy interlacing inequalities tell us that

$\displaystyle 0 \leq \lambda_i(A) - \lambda_k(M) \leq \lambda_i(A) - \lambda_k(A)$

for ${1 \leq k < i}$, and

$\displaystyle \lambda_i(A) - \lambda_{k+1}(A) \leq \lambda_i(A) - \lambda_k(M) < 0$

for ${i \leq k \leq n-1}$, so that the right-hand side of (1) lies between ${0}$ and ${1}$, which is of course consistent with (1) as ${v_i}$ is a unit vector. Thus the identity relates the coefficient sizes of an eigenvector with the extent to which the Cauchy interlacing inequalities are sharp.