野村数学研究所

\[ \left(\begin{array}{cccc} A_{1,1} & O & \cdots & O\\ O & A_{2,2} & \ddots & O\\ \vdots & \ddots & \ddots & \vdots\\ O & O & \cdots & A_{p,p} \end{array}\right)^{-1}=\left(\begin{array}{cccc} A_{1,1}^{-1} & O & \cdots & O\\ O & A_{2,2}^{-1} & \ddots & O\\ \vdots & \ddots & \ddots & \vdots\\ O & O & \cdots & A_{p,p}^{-1} \end{array}\right) \]

\[ \tr\left(\begin{array}{cccc} A_{1,1} & A_{1,2} & \cdots & A_{1,p}\\ A_{2,1} & A_{2,2} & \ddots & A_{2,p}\\ \vdots & \ddots & \ddots & \vdots\\ A_{p,1} & A_{p,2} & \cdots & A_{p,p} \end{array}\right)=\sum_{k=1}^{p}\tr\left(A_{k,k}\right) \]

\[ \left(\begin{array}{cccc} A_{11} & O & \cdots & O\\ O & A_{22} & \ddots & O\\ \vdots & \ddots & \ddots & \vdots\\ O & O & \cdots & A_{pp} \end{array}\right)^{k}=\left(\begin{array}{cccc} A_{11}^{k} & O & \cdots & O\\ O & A_{22}^{k} & \ddots & O\\ \vdots & \ddots & \ddots & \vdots\\ O & O & \cdots & A_{pp}^{k} \end{array}\right) \]

\[ \det\left(\begin{array}{cccc} A_{1,1} & O & \cdots & O\\ A_{1,2} & A_{2,2} & \ddots & O\\ \vdots & \ddots & \ddots & \vdots\\ A_{1,p} & A_{2,p} & \cdots & A_{p,p} \end{array}\right)=\prod_{k=1}^{p}\det\left(A_{k,k}\right) \]

\[ \left(\begin{array}{cc} A & B\\ O & D \end{array}\right)^{-1}=\left(\begin{array}{cc} A^{-1} & -A^{-1}BD^{-1}\\ O & D^{-1} \end{array}\right) \]

\[ \det\left(\begin{array}{cc} A & O\\ C & D \end{array}\right)=\det\left(A\right)\det\left(D\right) \]

\[ \left(\begin{array}{cc} A & B\\ C & D \end{array}\right)=\left(\begin{array}{cc} I & O\\ CA^{-1} & I \end{array}\right)\left(\begin{array}{cc} A & O\\ O & D-CA^{-1}B \end{array}\right)\left(\begin{array}{cc} I & A^{-1}B\\ O & I \end{array}\right) \]

\[ \left[AB\right]_{i,j}=\sum_{k=1}^{q}A_{i,k}B_{k,j} \]