dot

Dot product

Syntax

C = dot(A,B)

C = dot(A,B,dim)

Description

C = dot(A,B) returns the scalar dot product of A and B.

• If A and B are vectors, then they must have the same length.

• If A and B are matrices or multidimensional arrays, then they must have the same size. In this case, the dot function treats A and B as collections of vectors. The function calculates the dot product of corresponding vectors along the first array dimension whose size does not equal 1.

C = dot(A,B,dim) evaluates the dot product of A and B along dimension, dim. The dim input is a positive integer scalar.

Examples

Dot Product of Real Vectors

Create two simple, three-element vectors.

A = [4 -1 2];
B = [2 -2 -1];

Calculate the dot product of A and B.

C = dot(A,B)

C = 8

The result is 8 since

C = A(1)*B(1) + A(2)*B(2) + A(3)*B(3)

Dot Product of Complex Vectors

Create two complex vectors.

A = [1+i 1-i -1+i -1-i];
B = [3-4i 6-2i 1+2i 4+3i];

Calculate the dot product of A and B.

C = dot(A,B)

C = 1.0000 - 5.0000i

The result is a complex scalar since A and B are complex. In general, the dot product of two complex vectors is also complex. An exception is when you take the dot product of a complex vector with itself.

Find the inner product of A with itself.

D = dot(A,A)

D = 8

The result is a real scalar. The inner product of a vector with itself is related to the Euclidean length of the vector, norm(A).

Dot Product of Matrices

Create two matrices.

A = [1 2 3;4 5 6;7 8 9];
B = [9 8 7;6 5 4;3 2 1];

Find the dot product of A and B.

C = dot(A,B)

C = 1×3

    54    57    54

The result, C, contains three separate dot products. dot treats the columns of A and B as vectors and calculates the dot product of corresponding columns. So, for example, C(1) = 54 is the dot product of A(:,1) with B(:,1).

Find the dot product of A and B, treating the rows as vectors.

D = dot(A,B,2)

D = 3×1

    46
    73
    46

In this case, D(1) = 46 is the dot product of A(1,:) with B(1,:).

Dot Product of Multidimensional Arrays

Create two multidimensional arrays.

A = cat(3,[1 1;1 1],[2 3;4 5],[6 7;8 9])

A = 
A(:,:,1) =

     1     1
     1     1


A(:,:,2) =

     2     3
     4     5


A(:,:,3) =

     6     7
     8     9

B = cat(3,[2 2;2 2],[10 11;12 13],[14 15; 16 17])

B = 
B(:,:,1) =

     2     2
     2     2


B(:,:,2) =

    10    11
    12    13


B(:,:,3) =

    14    15
    16    17

Calculate the dot product of A and B along the third dimension (dim = 3).

C = dot(A,B,3)

C = 2×2

   106   140
   178   220

The result, C, contains four separate dot products. The first dot product, C(1,1) = 106, is equal to the dot product of A(1,1,:) with B(1,1,:).

Input Arguments

A,B — Input arrays

numeric arrays

Input arrays, specified as numeric arrays.

Data Types: single | double
Complex Number Support: Yes

dim — Dimension to operate along

positive integer scalar

Dimension to operate along, specified as a positive integer scalar. If no value is specified, the default is the first array dimension whose size does not equal 1.

Consider two 2-D input arrays, A and B:

• dot(A,B,1) treats the columns of A and B as vectors and returns the dot products of corresponding columns.

• dot(A,B,2) treats the rows of A and B as vectors and returns the dot products of corresponding rows.

dot returns conj(A).*B if dim is greater than ndims(A).

More About

Scalar Dot Product

The scalar dot product of two real vectors of length n is equal to

$$u·v={\displaystyle \sum _{i=1}^n{u}_i{v}_i=u_1{v}_1+u_2{v}_2+\mathrm{...}}+u_n{v}_n.$$

This relation is commutative for real vectors, such that dot(u,v) equals dot(v,u). If the dot product is equal to zero, then u and v are perpendicular.

For complex vectors, the dot product involves a complex conjugate. This ensures that the inner product of any vector with itself is real and positive definite.

$$u·v={\displaystyle \sum _{i=1}^n{\overline{u}}_i{v}_i}.$$

Unlike the relation for real vectors, the complex relation is not commutative, so dot(u,v) equals conj(dot(v,u)).

Algorithms

• When inputs A and B are real or complex vectors, the dot function treats them as column vectors and dot(A,B) is the same as sum(conj(A).*B).

• When the inputs are matrices or multidimensional arrays, the dim argument determines which dimension the sum function operates on. In this case, dot(A,B) is the same as sum(conj(A).*B,dim).

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

This function supports tall arrays with the limitation:

For the syntax dot(A,B), the arrays A and B must have the same size, even if they are vectors.

For more information, see Tall Arrays for Out-of-Memory Data.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Code generation does not support sparse matrix inputs for this function.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).