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Dot

or Dot

gives products of vectors, matrices, and tensors.

MORE INFORMATION

gives an explicit result when a and b are lists with appropriate dimensions. It contracts the last index in a with the first index in b.

Various applications of Dot:

	{a₁,a₂}.{b₁,b₂}	scalar product of vectors
	{a₁,a₂}.{{m₁₁,m₁₂},{m₂₁,m₂₂}}
		product of a vector and a matrix
	{{m₁₁,m₁₂},{m₂₁,m₂₂}}.{a₁,a₂}
		product of a matrix and a vector
	{{m₁₁,m₁₂},{m₂₁,m₂₂}}.{{n₁₁,n₁₂},{n₂₁,n₂₂}}
		product of two matrices

The result of applying Dot to two tensors and is the tensor . Applying Dot to a rank tensor and a rank tensor gives a rank tensor. »

Dot can be used on SparseArray objects, returning a SparseArray object when possible. »

When its arguments are not lists or sparse arrays, Dot remains unevaluated. It has the attribute Flat.

Basic Examples (3)

Scalar product of vectors:

Products of matrices and vectors:

Matrix product:

Scalar product of vectors:

Out[1]=

Products of matrices and vectors:

Out[1]=

Out[2]=

Out[3]=

Matrix product:

Out[1]=

and

are 5×5 random matrices of zeros and ones:

Use exact arithmetic to find the matrix product of

and

:

Use machine arithmetic:

Use higher-precision arithmetic:

Use SparseArray objects:

Compute the matrix product of random real and complex rectangular matrices:

Generalizations & Extensions (1)

Dot works for tensors:

The dimensions of the result are those of the input with the common dimension collapsed:

Any combination is allowed as long as products are done with a common dimension:

Applications (1)

A linear mapping

:

Get the matrix representation

for the linear mapping:

Apply the linear mapping to a vector:

Using the matrix with Dot is faster:

Properties & Relations (4)

is a 2×3×4 tensor and

is a 4×5 random matrix:

The result of applying Dot to two tensors

and

is the tensor

:

Applying Dot to a rank

tensor and a rank

tensor gives a rank

tensor.

is a random complex vector:

is given by

:

is a 3×3 matrix:

Compute the matrix product

:

This is the same as MatrixPower:

This is equivalent to composing the action of

on a vector three times:

Dot is a special case of Inner:

Possible Issues (2)

Dot effectively treats vectors multiplied from the right as column vectors:

Dot effectively treats vectors multiplied from the left as row vectors:

To get an outer product, you need to form the inputs as matrices:

Or you can use KroneckerProduct:

MatrixPower Cross Norm KroneckerProduct Inner Outer AffineTransform NonCommutativeMultiply VectorAngle Covariance

Vectors and Matrices

Multiplying Vectors and Matrices

NKS|Online (A New Kind of Science)

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