Understanding the Dot Product: Computing Dot Product of Matrices and Vectors

dot product of a matrix

The dot product of two matrices is only defined between two vectors

The dot product of two matrices is only defined between two vectors. Therefore, to compute the dot product of a matrix, we have two options:

1. Dot product of two matrices treated as vectors: We can reshape the matrices into column vectors and then compute the dot product between them. To reshape a matrix into a column vector, we simply “stack” the columns on top of each other. Let’s say we have two matrices A and B of dimensions m x n. We can reshape them into column vectors as follows:

A = [a₁₁ a₁₂ … a₁ₙ;
a₂₁ a₂₂ … a₂ₙ;
… … …;
aₘ₁ aₘ₂ … aₘₙ]

B = [b₁₁ b₁₂ … b₁ₙ;
b₂₁ b₂₂ … b₂ₙ;
… … …;
bₘ₁ bₘ₂ … bₘₙ]

The column vector representation of A will be [a₁₁, a₂₁, …, aₘ₁, a₁₂, a₂₂, …, aₘ₂, …, a₁ₙ, a₂ₙ, …, aₘₙ], and similarly for B.

Then, we can compute the dot product of A and B as the product of their column vectors:

A · B = [a₁₁, a₂₁, …, aₘ₁, a₁₂, a₂₂, …, aₘ₂, …, a₁ₙ, a₂ₙ, …, aₘₙ] · [b₁₁, b₂₁, …, bₘ₁, b₁₂, b₂₂, …, bₘ₂, …, b₁ₙ, b₂ₙ, …, bₘₙ]
= a₁₁ * b₁₁ + a₂₁ * b₂₁ + … + aₘ₁ * bₘ₁ + a₁₂ * b₁₂ + … + aₘₙ * bₘₙ

2. Dot product of a matrix and vector: We can also compute the dot product between a matrix and a vector. In this case, the matrix is treated as a collection of row vectors, and the dot product is computed between each row vector and the given vector. Let’s say we have a matrix A of dimensions m x n and a vector V of length n. We can compute the dot product between A and V as follows:

A · V = [a₁₁, a₁₂, …, a₁ₙ;
a₂₁, a₂₂, …, a₂ₙ;
… …;
aₘ₁, aₘ₂, …, aₘₙ] · [v₁, v₂, …, vₙ]^T
= [a₁₁ * v₁ + a₁₂ * v₂ + … + a₁ₙ * vₙ;
a₂₁ * v₁ + a₂₂ * v₂ + … + a₂ₙ * vₙ;
…;
aₘ₁ * v₁ + aₘ₂ * v₂ + … + aₘₙ * vₙ]

So, depending on the context, we can compute the dot product of a matrix by treating it as a vector (option 1) or by computing the dot product between its rows and a given vector (option 2).

More Answers: