matrix_multiplication

设 \(\oplus,\otimes\) 为二元运算，且 \(\oplus\) 满足交换律和结合律，\(\otimes\) 满足结合律，\(\otimes\) 对 \(\oplus\) 满足分配律。定义新的矩阵乘法 \(A_{n,m}\times B_{m,p}=C_{n,p}\)，其中 \(c_{i,j}=\bigoplus_{k=1}^{m}a_{i,k}\otimes b_{k,j}\)，那么这种矩阵乘法满足结合律。

update：简单的说就是环上定义的矩阵仍然成一个环。之前写的什么玩意。

证明：设 \(D_{n,q}=(A_{n,m}\times B_{m,p})\times C_{p,q}\)，\(E_{n,q}=A_{n,m}\times(B_{m,p}\times C_{p,q})\)，则有： \[ \begin{aligned} d_{i,j}&=\bigoplus_{k=1}^{p}((\bigoplus_{u=1}^{m}a_{i,u}\otimes b_{u,k})\otimes c_{k,j})\\ \end{aligned} \] 由于 \(\otimes\) 对 \(\oplus\) 满足分配律，\(\otimes\) 满足结合律，因此 \[ \begin{aligned} d_{i,j}&=\bigoplus_{k=1}^{p}\bigoplus_{u=1}^{m}((a_{i,u}\otimes b_{u,k})\otimes c_{k,j})\\ &=\bigoplus_{k=1}^{p}\bigoplus_{u=1}^{m}(a_{i,u}\otimes b_{u,k}\otimes c_{k,j})\\ \end{aligned} \] 同理 \[ \begin{aligned} e_{i,j}&=\bigoplus_{u=1}^{m}(a_{i,u}\otimes(\bigoplus_{k=1}^{p}b_{u,k}\otimes c_{k,j}))\\ &=\bigoplus_{u=1}^{m}\bigoplus_{k=1}^{p}(a_{i,u}\otimes(b_{u,k}\otimes c_{k,j}))\\ &=\bigoplus_{u=1}^{m}\bigoplus_{k=1}^{p}(a_{i,u}\otimes b_{u,k}\otimes c_{k,j})\\ &=\bigoplus_{k=1}^{p}\bigoplus_{u=1}^{m}(a_{i,u}\otimes b_{u,k}\otimes c_{k,j})\\ &=d_{i,j} \end{aligned} \] 因此 \((A_{n,m}\times B_{m,p})\times C_{p,q}=A_{n,m}\times(B_{m,p}\times C_{p,q})\)，故该种矩阵乘法满足结合律。\(\Box\)