[R] matrix cross product in R different from cross product in Matlab

[Duncan Murdoch]

Susanne Schmidt wrote:
> Hi all,
>
> I have been searching all sorts of documentation, reference cards, cheat 
> sheets but can't find why R's
> crossprod(A, B) which is identical to A%*%B
>   

If you read the help page ?crossprod, you'll see it is supposed to be 
identical to t(A) %*% B.
> does not produce the same as Matlabs
> cross(A, B)
> Supposedly both calculate the cross product, and say so, or where do I 
> go wrong?
>   

They are following different definitions for "cross product".  The R 
definition is the statistical one, e.g. what is described here:  
http://stattrek.com/matrix-algebra/sums-of-squares.aspx .  The one you 
describe below is the one from 3 dimensional vector geometry.  You can 
get that one in R in a few different packages, but it's not really used 
much in statistics, so it's not in base R.  (It's called xprod in RFOC 
and RSEIS, extprod3d in geometry.)

Duncan Murdoch

> R is only doing sums in the crossprod however, as indicated by
>
> (z <- crossprod(1:4))    # = sum(1 + 2^2 + 3^2 + 4^2) 
> in 
>
> http://127.0.0.1:13073/library/base/html/crossprod.html
>
> R's result of
> g <- rbind(c(1, 2, 3), c( 4, 5, 6))
> h <- rbind(c(4, 5, 6), c(1, 2, 3))
> c <-  crossprod(g, h)
> is
>
>  > c
>      [,1] [,2] [,3]
> [1,]    8   13   18
> [2,]   13   20   27
> [3,]   18   27   36
>
> because there are only sums involved, however, I cant work out ALL of 
> them for the 3D example, something like
>          [,1]                     [,2]                     [,3]
> [1,]   1*4+4*1                 ?                   ?            # #sum 
> of elementwise multiplication of first column of first matrix with first 
> column of second matrix    of second matrix
> [2,]   ?                            2*5+5*2      ?
> ? cant figure out how these were derived, definitely this is NOT 
> identical to e.g. http://en.wikipedia.org/wiki/Cross_product
>
> I can work it out for the tcrossprod
>  > c <- tcrossprod(g, h)
>  > c
>      [,1] [,2]
> [1,]   32   14
> [2,]   77   32
>
> because
>             [,1]                     [,2]
>  [1,]     1*4+2*5+3*6   1*1+2*2+3*3   #sum of elementwise multiplication 
> of first line of first matrix with first line of second matrix   sum of 
> elementwise multiplication of first line of first matrix with second 
> line of second matrix
>  [2,]     4*4+5*5+6*6   4*1+5*2+6*3   #sum of elementwise multiplication 
> of second line of first matrix with first line of second matrix   sum of 
> elementwise multiplication of second line of first matrix with second 
> line of second matrix
>
>
> outer product is something completely different again , no need to show 
> it here
>
>
> Matlab's result in contrast of
> g = [1 2 3 ; 4 5 6]
> h = [4 5 6; 1 2 3]
> c = cross(g,h)
>
> is
> c =
>
>     -3     6    -3
>      3    -6     3
>
>
> which is in accordance to
> http://en.wikipedia.org/wiki/Cross_product
> and any math text book
>
>  >> d = dot(g, h)
>
> d =
>
>     32    50    72
> which is of course something different again
>
> What exactly does crossprod mean in R? and why does R's cross product 
> not do the same as indicated in references for cross products?
>
> base::crossprod
> doesnt tell me much:
> function (x, y = NULL)
> .Internal(crossprod(x, y))
> <environment: namespace:base>
>
> Where exactly do I look up the core of the function on Windows ? sorry, 
> I HAVE been trying to find this for hours now
> Thanks in advance,
> Susanne
>   
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