BUILT-IN MATHEMATICA SYMBOL

Outer

gives the generalized outer product of the

, forming all possible combinations of the lowest-level elements in each of them, and feeding them as arguments to f.

Outer

treats as separate elements only sublists at level n in the

.

Outer

treats as separate elements only sublists at level

in the corresponding

.

MORE INFORMATION

Outer[Times, list₁, list₂] gives an outer product.

The result of applying Outer to the tensors and is the tensor with elements . Applying Outer to two tensors of ranks r and s gives a tensor of rank .

The heads of all must be the same, but need not necessarily be List. »

The need not necessarily be cuboidal arrays.

The specifications of levels must be positive integers, or Infinity.

If only a single level specification is given, it is assumed to apply to all the . If there are several , but fewer than the number of , the lowest-level elements in the remaining will be used.

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

EXAMPLES

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Basic Examples (2)

Outer product of vectors:

Outer product of matrices:

In[1]:=

Out[1]=

Outer product of vectors:

In[1]:=

Out[1]=

Outer product of matrices:

In[2]:=

Out[2]=

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

and leaves from

to depth

:

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: