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# Dot (.)

 a.b.c or Dot[a, b, c]gives products of vectors, matrices and tensors.
• gives an explicit result when and are lists with appropriate dimensions. It contracts the last index in with the first index in .
• 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.  »
• When its arguments are not lists or sparse arrays, Dot remains unevaluated. It has the attribute Flat.
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]=
 Scope   (2)
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:
 Applications   (1)
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 n tensor and a rank m 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 :
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 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: