Scalar product of vectors:

Products of matrices and vectors:

Matrix product:

a and b are 5×5 random matrices of zeros and ones:

Use exact arithmetic to find the matrix product of a and b:

Use machine arithmetic:

Use higher-precision arithmetic:

Use SparseArray objects:

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

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:

A linear mapping :

Get the matrix representation m for the linear mapping:

Apply the linear mapping to a vector:

Using the matrix with Dot is faster:

a is a 2×3×4 tensor and b 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:

v is a random complex vector:

Norm[v] is given by :

a is a 3×3 matrix:

Compute the matrix product a.a.a:

This is the same as MatrixPower:

This is equivalent to composing the action of a on a vector three times:

Dot is a special case of Inner:

Dot can be implemented as a combination of TensorProduct and TensorContract:

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:

Or Outer:

Dot is not a Hermitian inner product:

Use Conjugate in one argument to get a Hermitian inner product:

Check that the result coincides with the square of the norm of a: