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dot

Dot product

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Syntax

C = dot(A,B)
C = dot(A,B,dim)

Description

example

C = dot(A,B)returns thescalar dot productofAandB.

  • IfAandBare vectors, then they must have the same length.

  • IfAandBare matrices or multidimensional arrays, then they must have the same size. In this case, thedotfunction treatsAandBas collections of vectors. The function calculates the dot product of corresponding vectors along the first array dimension whose size does not equal 1.

example

C = dot(A,B,dim)evaluates the dot product ofAandBalong dimension,dim. Thediminput is a positive integer scalar.

Examples

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Create two simple, three-element vectors.

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

Calculate the dot product ofAandB.

C = dot(A,B)
C = 8

The result is8since

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

Open Live Script

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 ofAandB.

C = dot(A,B)
C = 1.0000 - 5.0000i

The result is a complex scalar sinceAandBare 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 ofAwith 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).

Open Live Script

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 ofAandB.

C = dot(A,B)
C =54 57 54

The result,C, contains three separate dot products.dot对待the columns ofAandB为向量and calculates the dot product of corresponding columns. So, for example,C(1) = 54is the dot product ofA(:,1)withB(:,1).

Find the dot product ofAandB, treating therows为向量.

D = dot(A,B,2)
D =46 73 46

In this case,D(1) = 46is the dot product ofA(1,:)withB(1,:).

Open Live Script

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 ofAandBalong the third dimension (dim = 3).

C = dot(A,B,3)
C =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 ofA(1,1,:)withB(1,1,:).

Input Arguments

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Input arrays, specified as numeric arrays.

Data Types:single|double
Complex Number Support:Yes

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,AandB:

  • dot(A,B,1)对待the columns ofAandB为向量and returns the dot products of corresponding columns.

  • dot(A,B,2)对待the rows ofAandB为向量and returns the dot products of corresponding rows.

dotreturnsconj(A).*Bifdimis greater thanndims(A).

More About

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Scalar Dot Product

The scalar dot product of two real vectors of lengthnis equal to

u · v = ∑ i = 1 n u i v i = u 1 v 1 + u 2 v 2 + ... + u n v n .

This relation is commutative for real vectors, such thatdot(u,v)equalsdot(v,u). If the dot product is equal to zero, thenuandvare 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 = ∑ i = 1 n u ¯ i v i .

Unlike the relation for real vectors, the complex relation is not commutative, sodot(u,v)equalsconj(dot(v,u)).

Algorithms

  • When inputsAandBare real or complex vectors, thedot对待他们是列向量和函数dot(A,B)is the same assum(conj(A).*B).

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

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

See Also

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Introduced before R2006a

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