総合研究大学院大学 先端学術院
統計科学コース
2020年1月実施 数理 第3問

[問1]

\begin{align} \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} &= \begin{pmatrix} A_{11} & O \\ A_{21} & Q_1 \end{pmatrix} \begin{pmatrix} I & Q_2 \\ O & I \end{pmatrix} \\ &= \begin{pmatrix} A_{11} & A_{11} Q_2 \\ A_{21} & A_{21} Q_2 + Q_1 \end{pmatrix} \end{align} であるから、 \begin{align} Q_2 &= A_{11}^{-1} A_{12} ,\\ Q_1 &= A_{22} - A_{21} Q_2 \\ &= A_{22} - A_{21} A_{11}^{-1} A_{12} \end{align}

[問2]

\begin{align} \begin{pmatrix} I & O \\ O & I \end{pmatrix} &= \begin{pmatrix} B_{11} & O \\ B_{21} & B_{22} \end{pmatrix} \begin{pmatrix} B_{11}^{-1} & O \\ Q_3 & B_{22}^{-1} \end{pmatrix} \\ &= \begin{pmatrix} I & O \\ B_{21} B_{11}^{-1} + B_{22} Q_3 & I \end{pmatrix} \end{align} であるから、 \begin{align} Q_3 &= - B_{22}^{-1} B_{21} B_{11}^{-1} \end{align}

[問3]

\begin{align} \begin{pmatrix} A^{11} & A^{12} \\ A^{21} & A^{22} \end{pmatrix} &= \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}^{-1} \\ &= \left[ \begin{pmatrix} A_{11} & O \\ A_{21} & Q_1 \end{pmatrix} \begin{pmatrix} I & Q_2 \\ O & I \end{pmatrix} \right]^{-1} &\left( \because \text{ [問1] } \right) \\ &= \begin{pmatrix} I & Q_2 \\ O & I \end{pmatrix}^{-1} \begin{pmatrix} A_{11} & O \\ A_{21} & Q_1 \end{pmatrix}^{-1} \\ &= \begin{pmatrix} I & - Q_2 \\ O & I \end{pmatrix} \begin{pmatrix} A_{11}^{-1} & O \\ Q_4 & Q_1^{-1} \end{pmatrix} &\left( Q_4 = - Q_1^{-1} A_{21} A_{11}^{-1} , \ \ \because \text{ [問2] } \right) \\ &= \begin{pmatrix} A_{11}^{-1} - Q_2 Q_4 & - Q_2 Q_1^{-1} \\ Q_4 & Q_1^{-1} \end{pmatrix} \\ &= \begin{pmatrix} A_{11}^{-1} + A_{11}^{-1} A_{12} Q_1^{-1} A_{21} A_{11}^{-1} & - A_{11}^{-1} A_{12} Q_1^{-1} \\ - Q_1 A_{21} A_{11}^{-1} & Q_1^{-1} \end{pmatrix} \\ &= \begin{pmatrix} A_{11}^{-1} + A_{11}^{-1} A_{12} \left( A_{22} - A_{21} A_{11}^{-1} A_{12} \right)^{-1} A_{21} A_{11}^{-1} & - A_{11}^{-1} A_{12} \left( A_{22} - A_{21} A_{11}^{-1} A_{12} \right)^{-1} \\ - \left( A_{22} - A_{21} A_{11}^{-1} A_{12} \right)^{-1} A_{21} A_{11}^{-1} & \left( A_{22} - A_{21} A_{11}^{-1} A_{12} \right)^{-1} \end{pmatrix} \end{align}

[問4]

\begin{align} & \left( A_{22} - A_{21} A_{11}^{-1} A_{12} \right) \left[ A_{22}^{-1} + A_{22}^{-1} A_{21} \left( A_{11} - A_{12} A_{22}^{-1} A_{21} \right)^{-1} A_{12} A_{22}^{-1} \right] \\ &= \cdots \\ &= I \end{align} より、 \begin{align} \left( A_{22} - A_{21} A_{11}^{-1} A_{12} \right)^{-1} = A_{22}^{-1} + A_{22}^{-1} A_{21} \left( A_{11} - A_{12} A_{22}^{-1} A_{21} \right)^{-1} A_{12} A_{22}^{-1} \end{align}