有关舒尔补的一些公式，主要是涉及 \(2 \times 2\) 的分块矩阵的求逆和行列式。

舒尔补

对于一个非奇异的分块矩阵，其逆矩阵可以写为（假设 \(\mathbf{A}\) 可逆） \[ \left[\begin{array}{c} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \\ \end{array}\right]^{-1} = \left[\begin{array}{c} \mathbf{A}^{-1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \\ \end{array}\right] + \left[\begin{array}{c} -\mathbf{A}^{-1}\mathbf{B} \\ \mathbf{I} \\ \end{array}\right](\mathbf{D-CA^{-1}B})^{-1}\left[\begin{array}{c} \mathbf{-CA^{-1}} & \mathbf{I} \\ \end{array}\right] \] 其中矩阵 \(\mathbf{D-CA^{-1}B}\) 称为 \(\mathbf{A}\) 的舒尔补 (Schur complement) 。

类似地，我们有 \[ \left[\begin{array}{c} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \\ \end{array}\right]^{-1} = \left[\begin{array}{c} \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{D}^{-1} \\ \end{array}\right] + \left[\begin{array}{c} \mathbf{I} \\ -\mathbf{D}^{-1}\mathbf{C} \\ \end{array}\right](\mathbf{A}-\mathbf{B D}^{-1} \mathbf{C})^{-1}\left[\begin{array}{c} \mathbf{I} & \mathbf{-BD^{-1}}\\ \end{array}\right] \] 其中矩阵 \(\mathbf{A}-\mathbf{B D}^{-1} \mathbf{C}\) 称为矩阵 \(\mathbf{D}\) 的舒尔补。

Marsaglia 和 Styan (1974a,b) 给出了两个重要公式，分别为（第二个式子可以理解为将 \(\mathbf{A}\) 替换成了 \(\mathbf{-A}\) ）。 \[ \left(\mathbf{D}-\mathbf{C A}^{-1} \mathbf{B}\right)^{-1}=\mathbf{D}^{-1}+\mathbf{D}^{-1} \mathbf{C}\left(\mathbf{A}-\mathbf{B D}^{-1} \mathbf{C}\right)^{-1} \mathbf{B D}^{-1}, \]

\[ \left(\mathbf{D}+\mathbf{C A}^{-1} \mathbf{B}\right)^{-1}=\mathbf{D}^{-1}-\mathbf{D}^{-1} \mathbf{C}\left(\mathbf{A}+\mathbf{B D}^{-1} \mathbf{C}\right)^{-1} \mathbf{B D}^{-1} \text {, } \]

证明就是用右手项乘以 \(\mathbf{D-CA^{-1}B}\) ，证明过程略。

\(\mathbf{D-CA^{-1}B}\) 的行列式推导如下，首先根据行列式性质，我们有 \[ \left[\begin{array}{c} \mathbf{R} & \mathbf{0} \\ \mathbf{X} & \mathbf{T} \\ \end{array}\right] = |\mathbf{R}||\mathbf{T}| = \left[\begin{array}{c} \mathbf{R}^{\prime} & \mathbf{X}^{\prime} \\ \mathbf{0} & \mathbf{T}^{\prime} \\ \end{array}\right] \] 因此我们有 \[ \begin{aligned} \operatorname{det}\left[\begin{array}{ll} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{array}\right] &=\operatorname{det}\left(\left[\begin{array}{cc} \mathbf{A} & \mathbf{O} \\ \mathbf{C} & \mathbf{D}-\mathbf{C A}^{-1} \mathbf{B} \end{array}\right]\left[\begin{array}{cc} \mathbf{I} & \mathbf{A}^{-1} \mathbf{B} \\ \mathbf{O} & \mathbf{I} \end{array}\right]\right) \\ &=|\mathbf{A}| |\mathbf{D}-\mathbf{C A}^{-1} \mathbf{B}| \end{aligned} \] 类似的，我们有 \[ \begin{aligned} \operatorname{det}\left[\begin{array}{ll} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{array}\right] &=\operatorname{det}\left(\left[\begin{array}{cc} \mathbf{\mathbf{A}-\mathbf{B D}^{-1} \mathbf{C}} & \mathbf{B} \\ \mathbf{0} & \mathbf{D} \end{array}\right]\left[\begin{array}{cc} \mathbf{I} & \mathbf{0} \\ \mathbf{D}^{-1} \mathbf{C} & \mathbf{I} \end{array}\right]\right) \\ &=|\mathbf{D}| |\mathbf{A}-\mathbf{B D}^{-1} \mathbf{C}| \end{aligned} \] 因此，我们得到 \[ |\mathbf{D}-\mathbf{C A}^{-1} \mathbf{B}| = (|\mathbf{D}|/|\mathbf{A}|) |\mathbf{A}-\mathbf{B D}^{-1} \mathbf{C}| \]

参考文献

1. Searle S R, Casella G, McCulloch C E. Variance components[M]. John Wiley & Sons, 2009.
2. Matsaglia G, PH Styan G. Equalities and inequalities for ranks of matrices[J]. Linear and multilinear Algebra, 1974, 2(3): 269-292.
3. Marsaglia G, Styan G P H. Rank conditions for generalized inverses of partitioned matrices[J]. Sankhyā: The Indian Journal of Statistics, Series A, 1974: 437-442.