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matrix_product [2013/09/17 23:00] nikolaj |
matrix_product [2013/09/17 23:01] nikolaj |
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| ==== Definition ==== | ==== Definition ==== | ||
| | @#88DDEE: $R$ ... ring | | | @#88DDEE: $R$ ... ring | | ||
| - | | @#88DDEE: $m,n,r\in \mathbb N$ | | + | | @#88DDEE: $m,n,k\in \mathbb N$ | |
| - | | @#FFBB00: $ *: \mathrm{Matrix}(m,n,R)\times \mathrm{Matrix}(n,r,R)\to \mathrm{Matrix}(m,r,R) $ | | + | | @#FFBB00: $ *: \mathrm{Matrix}(m,n,R)\times \mathrm{Matrix}(n,k,R)\to \mathrm{Matrix}(m,k,R) $ | |
| - | | @#FFBB00: $ (A*B)_{jk}:=\sum_{l=1}^m\ A_{jl}\cdot B_{lk} $ | | + | | @#FFBB00: $ (A*B)_{ij}:=\sum_{l=1}^m A_{il}\cdot B_{lj} $ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| - | For square matrices, the matrix product is associative. Even for general matrices, we still have $(A*B)*'C=A*''(B*'''C)$, where the four binary functions are the matrix products for the suitable dimensions. | + | For square matrices, the matrix product is associative. And also for general matrices, we still have $(A*B)*'C=A*''(B*'''C)$, where the four binary functions are the matrix products for the suitable dimensions. |
| ==== Parents ==== | ==== Parents ==== | ||
| === Subset of === | === Subset of === | ||
| [[Matrix]], [[Function]] | [[Matrix]], [[Function]] | ||
