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Outer

Outer[f, list1, list2, ...]
gives the generalized outer product of the listi, forming all possible combinations of the lowest-level 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.
  • 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 lowest-level elements in the remaining listi will be used.
EXAMPLESCLOSE ALL
Basic Examples   (2)
Outer product of vectors:
Outer product of matrices:
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Outer product of vectors:
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Outer product of matrices:
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Scope   (4)
Treat nested lists as rank-1 vectors of sublists:
Arrays can be ragged:
Outer product of SparseArray objects:
Generalizations & Extensions   (1)
The head need not be List:
Applications   (6)
Word combinations:
Function combinations:
Complete bipartite graph:
Lower-triangular matrix:
Generate all possible binary trees with nodes from f and leaves from e to depth n:
Apply a function on a tensor product grid:
Show a contour plot of the values and the grid:
Include coordinates:
Make a piecewise polynomial that interpolates the data:
Properties & Relations   (5)
The dimensions of the result are a concatenation of the dimensions of the inputs:
Distribute forms the same combinations of all elements, but in a flat structure:
KroneckerProduct is a flattened outer product of matrices:
Part effectively uses an outer product when given lists of parts at multiple levels:
Table can also make a generalized outer product from lists:
Possible Issues   (1)
If backgrounds are inconsistent, a generalized outer product may not be sparse:
You can convert it into a SparseArray by choosing a background:
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