高等代数(四)

§ 7 § 7 §7 分块乘法的初等变换及应用举例 将分块乘法与初等变换结合是矩阵运算中极重要的手段. 现将某个单位矩阵进行如下分块: ( E m O O E n ) . \left(\begin{array}{cc} \boldsymbol{E}_{m} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{E}_{n} \end{array}\right) . (Em​O​OEn​​). 对它进行两行 (列) 对换, 某一行 (列) 左乘 (右乘)一个矩阵 P \boldsymbol{P} P, 一行 (列) 加上另一 行 (列) 的 P \boldsymbol{P} P (矩阵) 倍数, 就可得到如下类型的一些矩阵: ( O E n E m O ) , ( P O O E n ) , ( E m O O P ) , ( E m P O E n ) , ( E m O P E n ) . \left(\begin{array}{cc} \boldsymbol{O} & \boldsymbol{E}_{n} \\ \boldsymbol{E}_{m} & O \end{array}\right),\left(\begin{array}{ll} \boldsymbol{P} & \boldsymbol{O} \\ O & E_{n} \end{array}\right),\left(\begin{array}{cc} \boldsymbol{E}_{m} & \boldsymbol{O} \\ O & P \end{array}\right),\left(\begin{array}{cc} \boldsymbol{E}_{m} & \boldsymbol{P} \\ O & E_{n} \end{array}\right),\left(\begin{array}{cc} \boldsymbol{E}_{m} & O \\ P & E_{n} \end{array}\right) . (OEm​​En​O​),(PO​OEn​​),(Em​O​OP​),(Em​O​PEn​​),(Em​P​OEn​​). 和初等矩阵与初等变换的关系一样, 用这些矩阵左乘任一个分块矩阵 ( A B C D ) , \left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{B} \\ \boldsymbol{C} & \mathrm{D} \end{array}\right), (AC​BD​), 只要分块乘法能够进行, 其结果就是对它进行相应的变换, 即 ( O E m E n O ) ( A B C D ) = ( C D A B ) , ( P O O E n ) ( A B C D ) = ( P A P B C D ) , ( E m O P E n ) ( A B C D ) = ( A B C + P A D + P B ) . \begin{array}{c} \left(\begin{array}{cc} \boldsymbol{O} & \boldsymbol{E}_{m} \\ E_{n} & O \end{array}\right)\left(\begin{array}{ll} \boldsymbol{A} & \boldsymbol{B} \\ \boldsymbol{C} & D \end{array}\right)=\left(\begin{array}{ll} \boldsymbol{C} & \boldsymbol{D} \\ \boldsymbol{A} & \boldsymbol{B} \end{array}\right), \\ \left(\begin{array}{ll} P & O \\ O & E_{n} \end{array}\right)\left(\begin{array}{ll} A & B \\ C & D \end{array}\right)=\left(\begin{array}{cc} P A & P B \\ C & D \end{array}\right), \\ \left(\begin{array}{cc} E_{m} & O \\ P & E_{n} \end{array}\right)\left(\begin{array}{ll} A & B \\ C & D \end{array}\right)=\left(\begin{array}{cc} A & B \\ C+P A & D+P B \end{array}\right) . \end{array} (OEn​​Em​O​)(AC​BD​)=(CA​DB​),(PO​OEn​​)(AC​BD​)=(PAC​PBD​),(Em​P​OEn​​)(AC​BD​)=(AC+PA​BD+PB​).​

同样, 用它们右乘任一矩阵, 进行分块乘法时也有相应的结果, 我们不写出了. 在 (3) 中, 适当选择 P P P, 可使 C + P A = O C+P A=O C+PA=O. 例如 A A A 可逆时, 选 P = − C A − 1 P=-C A^{-1} P=−CA−1, 则 C + P A = O C+P A=O C+PA=O.于是 (3) 的右端成为 ( A B O D − C A − 1 B ) . \left(\begin{array}{cc} A & B \\ O & D-C A^{-1} B \end{array}\right) . (AO​BD−CA