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# numpy.outer

numpy.outer(a, b, out=None)

  Compute the outer product of two vectors.

  Given two vectors a and b of length `M` and `N`, respectively, the outer product [1] is:

      [[a_0*b_0  a_0*b_1 ... a_0*b_{N-1} ]
       [a_1*b_0    .
       [ ...          .
       [a_{M-1}*b_0            a_{M-1}*b_{N-1} ]]

  Parameters:
    a(M,) array_like

      First input vector. Input is flattened if not already 1-dimensional.

    b(N,) array_like

      Second input vector. Input is flattened if not already 1-dimensional.

    out(M, N) ndarray, optional

      A location where the result is stored

  Returns:
    out(M, N) ndarray

      `out[i, j] = a[i] * b[j]`

  See also

  `inner`
  `einsum`

    `einsum('i,j->ij', a.ravel(), b.ravel())` is the equivalent.

  `ufunc.outer`

    A generalization to dimensions other than 1D and other operations. `np.multiply.outer(a.ravel(), b.ravel())` is the equivalent.

  `linalg.outer`

    An Array API compatible variation of `np.outer`, which accepts 1-dimensional inputs only.

  `tensordot`

    `np.tensordot(a.ravel(), b.ravel(), axes=((), ()))` is the equivalent.

  References

  [1]

  G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Baltimore, MD, Johns Hopkins University Press, 1996, pg. 8.

  Examples

  Make a (very coarse) grid for computing a Mandelbrot set:

      >>> import numpy as np
      >>> rl = np.outer(np.ones((5,)), np.linspace(-2, 2, 5))
      >>> rl
      array([[-2., -1.,  0.,  1.,  2.],
             [-2., -1.,  0.,  1.,  2.],
             [-2., -1.,  0.,  1.,  2.],
             [-2., -1.,  0.,  1.,  2.],
             [-2., -1.,  0.,  1.,  2.]])
      >>> im = np.outer(1j*np.linspace(2, -2, 5), np.ones((5,)))
      >>> im
      array([[0.+2.j, 0.+2.j, 0.+2.j, 0.+2.j, 0.+2.j],
             [0.+1.j, 0.+1.j, 0.+1.j, 0.+1.j, 0.+1.j],
             [0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
             [0.-1.j, 0.-1.j, 0.-1.j, 0.-1.j, 0.-1.j],
             [0.-2.j, 0.-2.j, 0.-2.j, 0.-2.j, 0.-2.j]])
      >>> grid = rl + im
      >>> grid
      array([[-2.+2.j, -1.+2.j,  0.+2.j,  1.+2.j,  2.+2.j],
             [-2.+1.j, -1.+1.j,  0.+1.j,  1.+1.j,  2.+1.j],
             [-2.+0.j, -1.+0.j,  0.+0.j,  1.+0.j,  2.+0.j],
             [-2.-1.j, -1.-1.j,  0.-1.j,  1.-1.j,  2.-1.j],
             [-2.-2.j, -1.-2.j,  0.-2.j,  1.-2.j,  2.-2.j]])

  An example using a “vector” of letters:

      >>> x = np.array(['a', 'b', 'c'], dtype=object)
      >>> np.outer(x, [1, 2, 3])
      array([['a', 'aa', 'aaa'],
             ['b', 'bb', 'bbb'],
             ['c', 'cc', 'ccc']], dtype=object)