12月3日，陶哲軒在其個人博客上更新了一篇文章，

https://terrytao.wordpress.com/2019/12/03/eigenvectors-from-eigenvalues-a-survey-of-a-basic-identity-in-linear-algebra/

說他與合作者在arXiv上更新了此前關于特征值的文章

Eigenvectors from Eigenvalues。

更新后，陶哲軒等將文章題目改為

Eigenvectors from Eigenvalues:?

a survey of a basic identity in linear algebra

即注明這篇文章是一篇綜述。

arxiv.org/abs/1908.03795

文章的要點是一個特征向量-特征值等式。

熟悉背景的網友們知道(參考關于陶哲軒《來自特征值的特征向量》)，8月10日，陶哲軒與合作者在線發表了一篇文章。合作起因也很傳奇，是物理學家發現了一個簡單而神奇的公式，即特征向量-特征值等式，請教陶哲軒，于是合作了文章(參考陶哲軒為之驚嘆的最新公式)。這個公式的特點是很簡單自然，但很少見。

文章發表后，11月份quanta發表文章對這個故事進行介紹(中文翻譯文章也出來，引起了很大反響)而廣為人知。美國華人丁教授還寫了篇文章《與陶哲軒“共舞”的一個周末 | 數學家發現紀實》。

作者們也因此獲得了許多反饋。

中文世界的故事可以用一些人喜歡的“反轉”來描述。因為有人很快發現這個結論其實很早前就有了，特別是北大一位教授所著中文書中也有。有人因此批評陶哲軒作為著名數學家，所得到的發現只不過是很早前就有人發現過了的，做出“大神不過爾爾”的評論。

現在陶哲軒等人經過與人(在線討論、私下交流、文獻引用等)交流研究，發現這個公式在數值線性代數、隨機矩陣、圖論等各個領域常被證明，也常被遺忘。此前人們一般認為這個公式最早出自1968的一篇文章。現在陶哲軒等研究發現最早的相關文獻可以追溯到1934年。后來不斷有證明，陶哲軒本人自己也用過相關公式。

下面是陶哲軒自己在其博客中關于這個故事的敘述，并且給出了一個樹圖——文獻中證明、引用該等式的歷史。

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 L?wner?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.

陶哲軒等人的文章介紹了相關背景，給出了7個不同的證明。相信這個優美的公式將來不會再被遺忘。陶哲軒與特征值特征向量等式的故事完美地描述了一個數學家做研究的歷程。非常令人喜歡。

數學結果被忘記被再次發現，是常見現象。陶哲軒等人先前不知道這個結果的存在，不必苛責。這使得我們想起一位學生的問題：很多中學幾何題可以用解析代數的辦法求解，還用幾何方法有意義嗎？實際上，不同的證明給出了不同的理解。例如，陶哲軒等人的再研究讓我們對這個等式有了新的認識。

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