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「Dodgson condensation from Schur complementation」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

The determinant of an matrix (with coefficients in an arbitrary field) obey many useful identities, starting of course with the fundamental multiplicativity for matrices . This multiplicativity can in turn be used to establish many further identities; in particular, as shown in this previous post , it implies the Schur determinant identity

whenever is an invertible matrix, is an matrix, is a matrix, and is a matrix. The matrix is known as the Schur complement of the block .

已知结果和反例

I only recently discovered that this identity in turn immediately implies what I always found to be a somewhat curious identity, namely the Dodgson condensation identity (also known as the Desnanot-Jacobi identity )

for any and matrix , where denotes the matrix formed from by removing the row and column, and similarly denotes the matrix formed from by removing the and rows and and columns. Thus for instance when we obtain

证明或构造的主线

for any scalars . (Charles Dodgson, better known by his pen name Lewis Caroll , is of course also known for writing “ Alice in Wonderland ” and “ Through the Looking Glass “.)

The derivation is not new; it is for instance noted explicitly in this paper of Brualdi and Schneider , though I do not know if this is the earliest place in the literature where it can be found. (EDIT: Apoorva Khare has pointed out to me that the original arguments of Dodgson can be interpreted as implicitly following this derivation.) 一个常见想法是 it is worth presenting the short derivation here, though.

阅读时建议盯住的点

Firstly, by swapping the first and rows, and similarly for the columns, it is easy to see that the Dodgson condensation identity is equivalent to the variant

where is an matrix, are column vectors, are row vectors, and are scalars. If is invertible, we may apply the Schur determinant identity repeatedly to conclude that

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