The Matrix Cookbook(译)

来源：互联网发布：真有外星人吗知乎编辑：程序博客网时间：2024/06/06 00:19

（其实也没什么好翻译的，全是公式，为了了解记忆，哈哈）

A - 1 表 示 A 的 逆 矩 阵; A T 表 示 A 的 转 置; A H 表 示 H e r m i t i a n 矩 阵 (A 的 共 轭 转 置 矩 阵 A * = = A) (A B) - 1 = B - 1 A - 1 (A B C . . .) - 1 = . . . C - 1 B - 1 A - 1 (A T) - 1 = (A - 1) T (A + B) T = A T + B T (A B) T = B T A T (A B C . . .) T = . . . C T B T A T (A H) - 1 = (A - 1) H (A + B) H = A H + B H (A B) H = B H A H (A B C . . .) H = . . . C H B H A H

1 迹（Trace）

* * * e i g (A) 表 示 A 的 特 征 值 * * * T r (A) = \sum i A i i T r (A) = \sum i λ i, λ i = e i g (A) T r (A) = T r (A T) T r (A B) = T r (B A) T r (A + B) = T r (A) + T r (B) T r (A B C) = T r (B C A) = T r (C A B) a T a = T r (a a T)

* * * 令 A \in R n \times n; e i g (A) 表 示 A 的 特 征 值 * * * d e t (A) = \prod i λ i, λ i = e i g (A) d e t (c A) = c n d e t (A) d e t (A T) = d e t (A) d e t (A B) = d e t (A) d e t (B) d e t (A - 1) = 1 d e t ( A ) d e t (A n) = d e t (A) n d e t (I + u v T) = I + u T v 对 于 n = 2 : d e t (I + A) = 1 + d e t (A) + T r (A) 对 于 n = 3 : d e t (I + A) = 1 + d e t (A) + T r (A) + 1 2 T r (A) 2 - 1 2 T r (A 2) 对 于 小 ε : d e t (I + ε A) ⋍ 1 + d e t (A) + ε T r (A) + 1 2 ε 2 T r (A) 2 - 1 2 ε 2 T r (A 2)

考 虑 A \in R 2 \times 2 : A = [A 11 A 12 A 21 A 22] 有 : d e t (A) = A 11 A 22 - A 12, A 21; T r (A) = A 11 + A 22 特 征 值 : λ 2 - λ \cdot T r (A) + d e t (A) = 0 逆; A - 1 = 1 d e t ( A ) [A 22 - A 12 - A 21 A 11]

慢慢来，后续ing。。。

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