# Wolfram Language & System 11.0 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# Outer

Outer[f,list1,list2,]
gives the generalized outer product of the listi, forming all possible combinations of the lowestlevel elements in each of them, and feeding them as arguments to f.

Outer[f,list1,list2,,n]
treats as separate elements only sublists at level n in the listi.

Outer[f,list1,list2,,n1,n2,]
treats as separate elements only sublists at level ni in the corresponding listi.

## DetailsDetails

• Outer[Times,list1,list2] gives an outer product.
• The result of applying Outer to the tensors Ti1i2...ir and Uj1j2...js is the tensor Vi1i2...irj1j2...js with elements f[Ti1i2...ir,Uj1j2...js]. Applying Outer to two tensors of ranks r and s gives a tensor of rank r+s.
• The heads of all listi must be the same, but need not necessarily be List. »
• The listi need not necessarily be cuboidal arrays.
• The specifications ni of levels must be positive integers, or Infinity.
• If only a single level specification is given, it is assumed to apply to all the listi. If there are several ni, but fewer than the number of listi, the lowestlevel elements in the remaining listi will be used.
• Outer can be used on SparseArray objects, returning a SparseArray object when possible. »

## ExamplesExamplesopen allclose all

### Basic Examples  (2)Basic Examples  (2)

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Outer product of vectors:

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Outer product of matrices:

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