| Re: Meaning of outer product [message #31427 is a reply to message #31426] |
Sat, 13 July 2002 11:17   |
James Kuyper
Messages: 425
Registered: March 2000
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Senior Member |
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Paul Sorenson wrote:
>
> Greetings,
>
> IDL documentation says: "Note - If A and B arguments are vectors, then C =
> MATRIX_MULTIPLY(A, B) is a matrix with C_ij = A_iB_j. Mathematically, this
> is equivalent to the outer product. . . ." But I'm having difficulty
> reconciling this with my understanding of outer product. . .
>
> c.x = a.y*b.z - a.z*b.y
> c.y = a.z*b.x - a.x*b.z
> c.z = a.x*b.y - a.y*b.x
>
> ... which yields a vector (c) instead of a 2D array. Can anyone shed some
> light on this?
The outer product you are referring to is also called the cross-product,
and is different from the outer product that the IDL documentation is
talking about. However, the two concepts are related. In general, the
outer product of two vectors could be written as:
OP = [[a.x*b*x, a.x*b.y, a.x*b.x],
[a.y*b*x, a.y*b.y, a.y*b.x],
[a.z*b*x, a.z*b.y, a.z*b.x]]
You can create such an outer product by matrix multiplication of column
vector by a row vector (matrix multiplication of a row vector by a
column vector gives their dot product).
While 'OP' has 9 different elements, they were calculated from only six
different original numbers. The entire information content of that array
can be summarized by forming two other matrices:
SYM = (OP + transpose(OP))/2 Symmetrized
ASYM = (OP - transpose(OP)) Anti-symmetrized
The ASYM matrix necessarily has 0 on it's diagonal elements, and the
off-diagonal elements come in pairs that are exactly equal in size, and
opposite in sign. Therefore, it can be described by exactly three
independent numbers. The cross-product is a special rearrangement of
those three numbers to create a pseudo-vector. It's a pseudo-vector
because it can be rotated the same way as a real vector, but remains
unchanged by mirror reflections. Mathematically, it's called the "dual"
of the anti-symmetrized outer product.
I'm not sure how the cross-produce ended up also being called the "outer
product", but it's probably because of this connection. For some
purposes, the three numbers in the cross-product represent all of the
useful information that's contained in the outer product.
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