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# numpy.outer

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

  Compute the outer product of two vectors.

  Given two vectors, `a = [a0, a1, ..., aM]` and `b = [b0, b1, ..., bN]`, the outer product [R55] is:

      [[a0*b0  a0*b1 ... a0*bN ]
       [a1*b0    .
       [ ...          .
       [aM*b0            aM*bN ]]

  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 New in version 1.9.0.
  Returns: | out : (M, N) ndarray `out[i, j] = a[i] * b[j]`

  See also

  `inner`, `einsum`

  References

  [R55] | (1, 2) : 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:

      >>> 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)