矩阵求逆引理（Matrix inversion lemma）推导

  矩阵求逆引理（Matrix inversion lemma）:

  现有矩阵$A$可以写为如下分块矩阵形式：

$$A=\begin{bmatrix} A_{11} & A_{12}\\ A_{21} & A_{22} \end{bmatrix}_{(m+n) \times (m+n)}$$

  矩阵$A$为$(m+n)$阶方阵，其中$A_{11}$为$n$阶非奇异方阵，$A_{22}$为$m$阶非奇异方阵。那么可以得到：$(A_{11} - A_{12}A_{22}^{-1}A_{21})$ 和 $(A_{22} - A_{21}A_{11}^{-1}A_{12})$都是非奇异矩阵。

  引理结论：

$$A^{-1} =\begin{bmatrix} A_{11}^{-1} + A_{11}^{-1}A_{12}(A_{22} - A_{21}A_{11}^{-1}A_{12})^{-1} A_{21}A_{11}^{-1} & - A_{11}^{-1}A_{12}(A_{22} - A_{21}A_{11}^{-1}A_{12})^{-1}\\ - (A_{22} - A_{21}A_{11}^{-1}A_{12})^{-1} A_{21}A_{11}^{-1} & (A_{22} - A_{21}A_{11}^{-1}A_{12})^{-1} \end{bmatrix}$$

A−1=[(A11−A12A−122A21)−1−A−122A21(A11−A12