Understanding the Dot Product of Matrices: A Comprehensive Guide

dot product of a matrix

The dot product of two matrices, typically denoted as A·B or A∙B, is a binary operation between two matrices that results in a scalar quantity

The dot product of two matrices, typically denoted as A·B or A∙B, is a binary operation between two matrices that results in a scalar quantity. However, the dot product of matrices is defined only when the two matrices have compatible dimensions.

To compute the dot product of two matrices, we first need to ensure that the number of columns in the first matrix is equal to the number of rows in the second matrix. Let’s consider two matrices A and B.

Matrix A has dimensions m × n: A = [aᵢⱼ], where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
Matrix B has dimensions n × p: B = [bⱼₖ], where 1 ≤ j ≤ n and 1 ≤ k ≤ p.

The dot product of A and B, denoted as C = A·B, will have dimensions m × p: C = [cᵢₖ], where 1 ≤ i ≤ m and 1 ≤ k ≤ p.

The elements of the resulting matrix C are computed by multiplying the corresponding elements of each row of A with the corresponding elements of each column of B, and summing up these products. Mathematically, it can be expressed as:

cᵢₖ = Σ(aᵢⱼ * bⱼₖ), for 1 ≤ j ≤ n.

The dot product operation is often used in linear algebra for various purposes, including matrix transformations, solving systems of linear equations, and calculating orthogonal projections.

It’s important to note that the dot product of matrices is different from the element-wise matrix multiplication, which is denoted by A * B or A ⊙ B. In element-wise multiplication, the matrices must have the same dimensions, and each element of the resulting matrix is obtained by multiplying the corresponding elements of the input matrices.

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