       Outer product in mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg51261] Outer product in mathematica
• From: Jonas Sourlier <aeroswiss at gmx.ch>
• Date: Mon, 11 Oct 2004 01:25:20 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

Hi there

In the lections at my University I have learnt that the outer product
of two Vectors is defined as follows:

(a)       x (b)       = 0
(a, b)    x (c, d)    = ad - bc
(a, b, c) x (d, e, f) = (bf - ce, cd - af, ae - bd)

For two four-dimensional Vectors the outer product produces a
six-dimensional Vector (handled as a skew-symmetric 4x4-matrix).
The general, axiomatic definition of the outer product says that it is

graduately anti-commutative: u x v = (-1)^(k+l) (v x u)
bilinear: (u + v) x w = u x w + v x w

for two vectors u, v and w with dimensions k, l, m.

Now, my question: The outer product seems to be implemented in
Mathematica with the function Outer. But whatever I've tried so far
with Outer I didn't manage to get the outer product of two vectors as
described above.

Outer[Times, {a,b,c},{d,e,f}]    produces the 3x3-Matrix

{{a d, a e, a f}, {b d, b e, b f}, {c d, c e, c f}}

How can I calculate the outer product of two vectors with Mathematica?

Thank a lot for helping me!
Jonas

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