陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「STOC talk: The condition number of a randomly perturbed matrix」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

This week I am in San Diego for the 39th ACM Symposium for the Theory of Computing (STOC). Today I presented my work with Van Vu on the condition number of randomly perturbed matrices, which was the subject of an earlier post on this blog . For this short talk (20 minutes), Van and I prepared some slides ; of course, in such a short time frame one cannot hope to discuss many of the details of the result, but one can at least convey the statement of the result and a brief sket

One late update (which didn’t make it onto the slides ): last week, an alternate proof of some cases of our main result (together with some further generalisations and other results, in particular a circular law for the eigenvalues of discrete random matrices) was obtained by Pan and Zhou , using earlier arguments by Rudelson and Vershynin .

已知结果和反例

[ Update , June 16: It was pointed out to me that the Pan-Zhou result only recovers our result in the case when the unperturbed matrix has a spectral norm of (our result assumes that the unperturbed matrix has polynomial size). Slides also updated.]

证明或构造的主线

先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。

阅读时建议盯住的点

先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源