dot product of a matrix
must have (m n) (n y)multiply row by column
The dot product of two matrices involves performing element-wise multiplication and summation of the resulting products.
Let’s say we have two matrices, A and B, and we want to find their dot product. The dot product of A and B, denoted as A⋅B, is defined as follows:
A⋅B = sum(Aij * Bij)
where i represents the row index, j represents the column index, and the sum is over all i and j values.
In order for the dot product to be defined, the number of columns in matrix A must equal the number of rows in matrix B. If A has size m x n and B has size n x p, then the resulting dot product will have dimensions m x p.
To find the dot product, we perform element-wise multiplication of the corresponding elements in each row of matrix A with each column of matrix B, and then sum the resulting products. This can be represented mathematically as follows:
A⋅B = [A11 A12 … A1n] ⋅ [B11 B12 … B1p]
[A21 A22 … A2n] ⋅ [B21 B22 … B2p]
[… … … …] [… … … …]
[Am1 Am2 … Amn] ⋅ [Bm1 Bm2 … Bmp]
= [sum(A11*B11 + A12*B21 + … + A1n*Bn1) sum(A11*B12 + A12*B22 + … + A1n*Bn2) … sum(A11*B1p + A12*B2p + … + A1n*Bnp)]
[sum(A21*B11 + A22*B21 + … + A2n*Bn1) sum(A21*B12 + A22*B22 + … + A2n*Bn2) … sum(A21*B1p + A22*B2p + … + A2n*Bnp)]
[… … … … ]
[sum(Am1*B11 + Am2*B21 + … + Amn*Bn1) sum(Am1*B12 + Am2*B22 + … + Amn*Bn2) … sum(Am1*B1p + Am2*B2p + … + Amn*Bnp)]
Therefore, the dot product of matrices A and B is a new matrix whose elements are the sum of the products of the corresponding elements in the rows and columns of A and B.
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