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Wayne Jones

2004-Oct-04 14:10 UTC

[R] Strange Matrix Multiplication Behaviour

Hi there fellow R-users, 

Im seeing some strange behaviour when I multiply a vector by a matrix

Here is my script: 
> tr        1         2         3         4         5         6 
0.2217903 0.1560525 0.1487908 0.1671354 0.1590643 0.1471667
> 
> ex1           a         b          c          d          e          f
1  0.2309579 -3.279045 -0.6694697 -1.1024404  0.2303928 -1.5527404
2 -0.2865357 -2.519789 -0.1138712 -0.3571832 -2.6727913 -0.3296945
> is.vector(tr)[1] TRUE
> is.data.frame(ex1)[1] TRUE
> tr* ex1[1,]           a          b           c          d          e          f
1 0.05122423 -0.5117031 -0.09961092 -0.1842569 0.03664727 -0.2285117
> (tr* ex1)[1,]           a          b          c          d          e          f
1 0.05122423 -0.4878917 -0.1064887 -0.2445106 0.03428033 -0.2469855


Notice that the output from tr * ex[1,] is different from (tr* ex1)[1,]
Especially element number 4. 

I would naturally expect both vectors to be equivalent. 


Can anyone shed any light on my predicament??


> version         _              
platform i386-pc-mingw32
arch     i386           
os       mingw32        
system   i386, mingw32  
status                  
major    1              
minor    9.1            
year     2004           
month    06             
day      21             
language R              



Regards

Wayne Jones






KSS Ltd
Seventh Floor  St James's Buildings  79 Oxford Street  Manchester  M1 6SS 
England
Company Registration Number 2800886
Tel: +44 (0) 161 228 0040	Fax: +44 (0) 161 236 6305
mailto:kssg@kssg.com		http://www.kssg.com


The information in this Internet email is confidential and m...{{dropped}}

Liaw, Andy

2004-Oct-04 14:48 UTC

head link

[R] Strange Matrix Multiplication Behaviour

What you need to realize is that R does vectorized computation in `Fortran'
order.  Here's a perhaps more illuminating example:
> tr <- rep(1:2, 3)
> tr
[1] 1 2 1 2 1 2> ex1 <- as.data.frame(matrix(1:12, 2, 6))
> ex1  V1 V2 V3 V4 V5 V6
1  1  3  5  7  9 11
2  2  4  6  8 10 12> tr * ex1[1,]  V1 V2 V3 V4 V5 V6
1  1  6  5 14  9 22> (tr * ex1)[1,]  V1 V2 V3 V4 V5 V6
1  1  3  5  7  9 11

When you do (tr * ex1)[1,], R multiplies tr and ex1 element-by-element,
stacking _columns_ of ex1 to match tr, and recyle the elements of tr to
match the length of the `stacked' ex1.

Andy
> From: Wayne Jones
> 
> Hi there fellow R-users, 
> 
> Im seeing some strange behaviour when I multiply a vector by a matrix
> 
> Here is my script: 
> 
> > tr
>         1         2         3         4         5         6 
> 0.2217903 0.1560525 0.1487908 0.1671354 0.1590643 0.1471667 
> > 
> > ex1
>            a         b          c          d          e          f
> 1  0.2309579 -3.279045 -0.6694697 -1.1024404  0.2303928 -1.5527404
> 2 -0.2865357 -2.519789 -0.1138712 -0.3571832 -2.6727913 -0.3296945
> 
> > is.vector(tr)
> [1] TRUE
> 
> > is.data.frame(ex1)
> [1] TRUE
> 
> > tr* ex1[1,]
>            a          b           c          d          e          f
> 1 0.05122423 -0.5117031 -0.09961092 -0.1842569 0.03664727 -0.2285117
> 
> > (tr* ex1)[1,]
>            a          b          c          d          e          f
> 1 0.05122423 -0.4878917 -0.1064887 -0.2445106 0.03428033 -0.2469855
> 
> 
> Notice that the output from tr * ex[1,] is different from 
> (tr* ex1)[1,]
> Especially element number 4. 
> 
> I would naturally expect both vectors to be equivalent. 
> 
> 
> Can anyone shed any light on my predicament??
> 
> 
> 
> > version
>          _              
> platform i386-pc-mingw32
> arch     i386           
> os       mingw32        
> system   i386, mingw32  
> status                  
> major    1              
> minor    9.1            
> year     2004           
> month    06             
> day      21             
> language R              
> 
> 
> 
> Regards
> 
> Wayne Jones
> 
> 
> 
> 
> 
> 
> KSS Ltd
> Seventh Floor  St James's Buildings  79 Oxford Street  
> Manchester  M1 6SS  England
> Company Registration Number 2800886
> Tel: +44 (0) 161 228 0040	Fax: +44 (0) 161 236 6305
> mailto:kssg at kssg.com		http://www.kssg.com
> 
> 
> The information in this Internet email is confidential and 
> m...{{dropped}}
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide! 
> http://www.R-project.org/posting-guide.html
> 
>

Huntsinger, Reid

2004-Oct-04 14:58 UTC

head link

[R] Strange Matrix Multiplication Behaviour

I think this is as intended. When you do elementwise multiplication of a
vector by a matrix R there are three important points to keep in mind: 

1) element-wise multiplication treats things as vectors
2) when you treat a matrix as a vector you get the _columns_ stacked one on
top of another. 
3) R recycles elements of the shorter vector

So tr*(ex[1,]) is the same as multiplying _row_ 1 by the elements of tr, but
(tr*ex)[1,] multiplies the first _column_ of ex by the first 2 elements of
tr, then the second column by the second two, then the third column by the
third two, then recycles for columns 4, 5 and 6. Then the [1,] takes the
first row of the result, so you (tr[1] tr[3] tr[5] tr[1] tr[3] tr[5]) times
the first row. Which looks like what you have.

Reid Huntsinger

-----Original Message-----
From: r-help-bounces at stat.math.ethz.ch
[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Wayne Jones
Sent: Monday, October 04, 2004 10:11 AM
To: R help list
Subject: [R] Strange Matrix Multiplication Behaviour



Hi there fellow R-users, 

Im seeing some strange behaviour when I multiply a vector by a matrix

Here is my script: 
> tr        1         2         3         4         5         6 
0.2217903 0.1560525 0.1487908 0.1671354 0.1590643 0.1471667
> 
> ex1           a         b          c          d          e          f
1  0.2309579 -3.279045 -0.6694697 -1.1024404  0.2303928 -1.5527404
2 -0.2865357 -2.519789 -0.1138712 -0.3571832 -2.6727913 -0.3296945
> is.vector(tr)[1] TRUE
> is.data.frame(ex1)[1] TRUE
> tr* ex1[1,]           a          b           c          d          e          f
1 0.05122423 -0.5117031 -0.09961092 -0.1842569 0.03664727 -0.2285117
> (tr* ex1)[1,]           a          b          c          d          e          f
1 0.05122423 -0.4878917 -0.1064887 -0.2445106 0.03428033 -0.2469855


Notice that the output from tr * ex[1,] is different from (tr* ex1)[1,]
Especially element number 4. 

I would naturally expect both vectors to be equivalent. 


Can anyone shed any light on my predicament??


> version         _              
platform i386-pc-mingw32
arch     i386           
os       mingw32        
system   i386, mingw32  
status                  
major    1              
minor    9.1            
year     2004           
month    06             
day      21             
language R              



Regards

Wayne Jones






KSS Ltd
Seventh Floor  St James's Buildings  79 Oxford Street  Manchester  M1 6SS
England
Company Registration Number 2800886
Tel: +44 (0) 161 228 0040	Fax: +44 (0) 161 236 6305
mailto:kssg at kssg.com		http://www.kssg.com


The information in this Internet email is confidential and m...{{dropped}}

______________________________________________
R-help at stat.math.ethz.ch mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide!
http://www.R-project.org/posting-guide.html

Prof Brian Ripley

2004-Oct-04 15:04 UTC

head link

[R] Strange Matrix Multiplication Behaviour

This is *not* matrix multiplication, which uses %*% in R.
Might that help explain your confusion?

The difference is explained in `An Introduction to R'.

On Mon, 4 Oct 2004, Wayne Jones wrote:
> 
> Hi there fellow R-users, 
> 
> Im seeing some strange behaviour when I multiply a vector by a matrix
> 
> Here is my script: 
> 
> > tr
>         1         2         3         4         5         6 
> 0.2217903 0.1560525 0.1487908 0.1671354 0.1590643 0.1471667 
> > 
> > ex1
>            a         b          c          d          e          f
> 1  0.2309579 -3.279045 -0.6694697 -1.1024404  0.2303928 -1.5527404
> 2 -0.2865357 -2.519789 -0.1138712 -0.3571832 -2.6727913 -0.3296945
> 
> > is.vector(tr)
> [1] TRUE
> 
> > is.data.frame(ex1)
> [1] TRUE
> 
> > tr* ex1[1,]
>            a          b           c          d          e          f
> 1 0.05122423 -0.5117031 -0.09961092 -0.1842569 0.03664727 -0.2285117
> 
> > (tr* ex1)[1,]
>            a          b          c          d          e          f
> 1 0.05122423 -0.4878917 -0.1064887 -0.2445106 0.03428033 -0.2469855
> 
> 
> Notice that the output from tr * ex[1,] is different from (tr* ex1)[1,]
> Especially element number 4. 
> 
> I would naturally expect both vectors to be equivalent. 
Why?  The second multiplies the elements of ex by tr, in column-major 
order, and what is `natural' about that?

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

Arne Henningsen

2004-Oct-04 15:09 UTC

head link

[R] Strange Matrix Multiplication Behaviour

Hi Wayne,

are you aware that you are NOT doing matrix multiplication? A matrix 
multiplication is done by %*%, e.g.> tr %*% t(as.matrix(ex1)
Or do you want a normal scalar multiplication with the elements of each row of 
ex1 to be multiplied with the corresponding element of tr? However, R does 
the multiplication not row-wise, but column-wise. Thus, the results are not 
as you expected. To do a row-wise multiplication you can transform the 
data.frame:
> tr <- as.vector(10*c(1:6))
> tr
[1] 10 20 30 40 50 60> ex1 <- as.data.frame(matrix(1:12,nrow=2,byrow=TRUE))
> ex1  V1 V2 V3 V4 V5 V6
1  1  2  3  4  5  6
2  7  8  9 10 11 12> tr * ex1   V1  V2  V3  V4  V5  V6
1  10  60 150  40 150 300
2 140 320 540 200 440 720> t( tr * t(ex1) )  V1  V2  V3  V4  V5  V6
1 10  40  90 160 250 360
2 70 160 270 400 550 720> t( tr * t(ex1) )[1,] V1  V2  V3  V4  V5  V6
 10  40  90 160 250 360> t( tr * t(ex1[1,]) )  V1 V2 V3  V4  V5  V6
1 10 40 90 160 250 360

I hope that this makes it clear.

Best wishes,
Arne


On Monday 04 October 2004 16:10, Wayne Jones wrote:> Hi there fellow R-users,
>
> Im seeing some strange behaviour when I multiply a vector by a matrix
>
> Here is my script:
> > tr
>
>         1         2         3         4         5         6
> 0.2217903 0.1560525 0.1487908 0.1671354 0.1590643 0.1471667
>
> > ex1
>
>            a         b          c          d          e          f
> 1  0.2309579 -3.279045 -0.6694697 -1.1024404  0.2303928 -1.5527404
> 2 -0.2865357 -2.519789 -0.1138712 -0.3571832 -2.6727913 -0.3296945
>
> > is.vector(tr)
>
> [1] TRUE
>
> > is.data.frame(ex1)
>
> [1] TRUE
>
> > tr* ex1[1,]
>
>            a          b           c          d          e          f
> 1 0.05122423 -0.5117031 -0.09961092 -0.1842569 0.03664727 -0.2285117
>
> > (tr* ex1)[1,]
>
>            a          b          c          d          e          f
> 1 0.05122423 -0.4878917 -0.1064887 -0.2445106 0.03428033 -0.2469855
>
>
> Notice that the output from tr * ex[1,] is different from (tr* ex1)[1,]
> Especially element number 4.
>
> I would naturally expect both vectors to be equivalent.
>
>
> Can anyone shed any light on my predicament??
>
> > version
>
>          _
> platform i386-pc-mingw32
> arch     i386
> os       mingw32
> system   i386, mingw32
> status
> major    1
> minor    9.1
> year     2004
> month    06
> day      21
> language R
>
>
>
> Regards
>
> Wayne Jones
>
>
>
>
>
>
> KSS Ltd
> Seventh Floor  St James's Buildings  79 Oxford Street  Manchester  M1
6SS
> England Company Registration Number 2800886
> Tel: +44 (0) 161 228 0040	Fax: +44 (0) 161 236 6305
> mailto:kssg at kssg.com		http://www.kssg.com
>
>
> The information in this Internet email is confidential and m...{{dropped}}
>
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide!
> http://www.R-project.org/posting-guide.html
-- 
Arne Henningsen
Department of Agricultural Economics
University of Kiel
Olshausenstr. 40
D-24098 Kiel (Germany)
Tel: +49-431-880 4445
Fax: +49-431-880 1397
ahenningsen at agric-econ.uni-kiel.de
http://www.uni-kiel.de/agrarpol/ahenningsen/

Wayne Jones

2004-Oct-04 15:28 UTC

head link

[R] Strange Matrix Multiplication Behaviour

Thanks for your help folks. 

I was not aware of the column wise nature of R multiplication. 

regards

Wayne


-----Original Message-----
From: Arne Henningsen [mailto:ahenningsen@email.uni-kiel.de] 
Sent: 04 October 2004 16:09
To: r-help@stat.math.ethz.ch
Cc: Wayne Jones
Subject: Re: [R] Strange Matrix Multiplication Behaviour

Hi Wayne,

are you aware that you are NOT doing matrix multiplication? A matrix 
multiplication is done by %*%, e.g.> tr %*% t(as.matrix(ex1)
Or do you want a normal scalar multiplication with the elements of each row
of 
ex1 to be multiplied with the corresponding element of tr? However, R does 
the multiplication not row-wise, but column-wise. Thus, the results are not 
as you expected. To do a row-wise multiplication you can transform the 
data.frame:
> tr <- as.vector(10*c(1:6))
> tr
[1] 10 20 30 40 50 60> ex1 <- as.data.frame(matrix(1:12,nrow=2,byrow=TRUE))
> ex1  V1 V2 V3 V4 V5 V6
1  1  2  3  4  5  6
2  7  8  9 10 11 12> tr * ex1   V1  V2  V3  V4  V5  V6
1  10  60 150  40 150 300
2 140 320 540 200 440 720> t( tr * t(ex1) )  V1  V2  V3  V4  V5  V6
1 10  40  90 160 250 360
2 70 160 270 400 550 720> t( tr * t(ex1) )[1,] V1  V2  V3  V4  V5  V6
 10  40  90 160 250 360> t( tr * t(ex1[1,]) )  V1 V2 V3  V4  V5  V6
1 10 40 90 160 250 360

I hope that this makes it clear.

Best wishes,
Arne


On Monday 04 October 2004 16:10, Wayne Jones wrote:> Hi there fellow R-users,
>
> Im seeing some strange behaviour when I multiply a vector by a matrix
>
> Here is my script:
> > tr
>
>         1         2         3         4         5         6
> 0.2217903 0.1560525 0.1487908 0.1671354 0.1590643 0.1471667
>
> > ex1
>
>            a         b          c          d          e          f
> 1  0.2309579 -3.279045 -0.6694697 -1.1024404  0.2303928 -1.5527404
> 2 -0.2865357 -2.519789 -0.1138712 -0.3571832 -2.6727913 -0.3296945
>
> > is.vector(tr)
>
> [1] TRUE
>
> > is.data.frame(ex1)
>
> [1] TRUE
>
> > tr* ex1[1,]
>
>            a          b           c          d          e          f
> 1 0.05122423 -0.5117031 -0.09961092 -0.1842569 0.03664727 -0.2285117
>
> > (tr* ex1)[1,]
>
>            a          b          c          d          e          f
> 1 0.05122423 -0.4878917 -0.1064887 -0.2445106 0.03428033 -0.2469855
>
>
> Notice that the output from tr * ex[1,] is different from (tr* ex1)[1,]
> Especially element number 4.
>
> I would naturally expect both vectors to be equivalent.
>
>
> Can anyone shed any light on my predicament??
>
> > version
>
>          _
> platform i386-pc-mingw32
> arch     i386
> os       mingw32
> system   i386, mingw32
> status
> major    1
> minor    9.1
> year     2004
> month    06
> day      21
> language R
>
>
>
> Regards
>
> Wayne Jones
>
>
>
>
>
>
> KSS Ltd
> Seventh Floor  St James's Buildings  79 Oxford Street  Manchester  M1
6SS
> England Company Registration Number 2800886
> Tel: +44 (0) 161 228 0040	Fax: +44 (0) 161 236 6305
> mailto:kssg@kssg.com		http://www.kssg.com
>
>
> The information in this Internet email is confidential and m...{{dropped}}
>
> ______________________________________________
> R-help@stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide!
> http://www.R-project.org/posting-guide.html
-- 
Arne Henningsen
Department of Agricultural Economics
University of Kiel
Olshausenstr. 40
D-24098 Kiel (Germany)
Tel: +49-431-880 4445
Fax: +49-431-880 1397
ahenningsen@agric-econ.uni-kiel.de
http://www.uni-kiel.de/agrarpol/ahenningsen/

	[[alternative HTML version deleted]]

Gavin Simpson

2004-Oct-04 15:34 UTC

head link

[R] Strange Matrix Multiplication Behaviour

Wayne Jones wrote:> Hi there fellow R-users, 
> 
> Im seeing some strange behaviour when I multiply a vector by a matrix
> 
> Here is my script: 
> 
"*" does array or element-wise multiplication. %*% is Matrix
multiplication.

In the first case, you are multiplying tr by the first row of ex1

 > tr * ex1[1, ]
            a          b           c          d          e          f
1 0.05122422 -0.5117032 -0.09961093 -0.1842568 0.03664727 -0.2285117

In the second, you are extracting the first row of the result of 
multiplying tr by ex1, which as we see below returns a 2x6 matrix:

 > tr * ex1
             a          b           c           d           e           f
1  0.05122422 -0.4878917 -0.10648873 -0.24451059  0.03428033 -0.24698556
2 -0.04471461 -0.4211459 -0.01675805 -0.05573933 -0.44671804 -0.04852005
 > (tr * ex1)[1, ]
            a          b          c          d          e          f
1 0.05122422 -0.4878917 -0.1064887 -0.2445106 0.03428033 -0.2469856

Note that tr gets recycled in the tr * ex1 and (tr * ex1)[1, ] as tr is 
not as long as ex1.

Element 1b in (tr * ex1)[1, ] is formed by multiplying element 1b of ex1 
(-3.279045) by the third element of the vector tr (0.1487908):

 > tr[3] * ex1[1,2]
[1] -0.4878917

the second value of the vector tr was used to multiply against element 
2a in matrix ex1:

but in the tr * ex1[1, ] case tr and ex1[1, ] both contain 6 elements 
and the element 1b in the results is formed by multiplying element 1b of 
ex1[1, ] (-3.279045) by the second element of vector tr (0.1560525)

 > tr[2] * ex1[1,2]
[1] -0.5117032
> 
> Notice that the output from tr * ex[1,] is different from (tr* ex1)[1,]
> Especially element number 4. 
The difference is due to recycling of tr and of where you are doing your 
sub setting
> I would naturally expect both vectors to be equivalent. 
> 
Hopefully my longwinded explanation helps.

Gav
-- 
%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%
Gavin Simpson                     [T] +44 (0)20 7679 5522
ENSIS Research Fellow             [F] +44 (0)20 7679 7565
ENSIS Ltd. & ECRC                 [E] gavin.simpson at ucl.ac.uk
UCL Department of Geography       [W] http://www.ucl.ac.uk/~ucfagls/cv/
26 Bedford Way                    [W] http://www.ucl.ac.uk/~ucfagls/
London.  WC1H 0AP.
%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%~%

Arne Henningsen

2004-Oct-04 15:35 UTC

head link

[R] Strange Matrix Multiplication Behaviour

Hi Wayne,

are you aware that you are NOT doing matrix multiplication? A matrix 
multiplication is done by %*%, e.g.> tr %*% t(as.matrix(ex1)
Or do you want a normal scalar multiplication with the elements of each row of 
ex1 to be multiplied with the corresponding element of tr? However, R does 
the multiplication not row-wise, but column-wise. Thus, the results are not 
as you expected. To do a row-wise multiplication you can transform the 
data.frame:
> tr <- as.vector(10*c(1:6))
> tr
[1] 10 20 30 40 50 60> ex1 <- as.data.frame(matrix(1:12,nrow=2,byrow=TRUE))
> ex1  V1 V2 V3 V4 V5 V6
1  1  2  3  4  5  6
2  7  8  9 10 11 12> tr * ex1   V1  V2  V3  V4  V5  V6
1  10  60 150  40 150 300
2 140 320 540 200 440 720> t( tr * t(ex1) )  V1  V2  V3  V4  V5  V6
1 10  40  90 160 250 360
2 70 160 270 400 550 720> t( tr * t(ex1) )[1,] V1  V2  V3  V4  V5  V6
 10  40  90 160 250 360> t( tr * t(ex1[1,]) )  V1 V2 V3  V4  V5  V6
1 10 40 90 160 250 360

I hope that this makes it clear.

Best wishes,
Arne


On Monday 04 October 2004 16:10, Wayne Jones wrote:> Hi there fellow R-users,
>
> Im seeing some strange behaviour when I multiply a vector by a matrix
>
> Here is my script:
> > tr
>
>         1         2         3         4         5         6
> 0.2217903 0.1560525 0.1487908 0.1671354 0.1590643 0.1471667
>
> > ex1
>
>            a         b          c          d          e          f
> 1  0.2309579 -3.279045 -0.6694697 -1.1024404  0.2303928 -1.5527404
> 2 -0.2865357 -2.519789 -0.1138712 -0.3571832 -2.6727913 -0.3296945
>
> > is.vector(tr)
>
> [1] TRUE
>
> > is.data.frame(ex1)
>
> [1] TRUE
>
> > tr* ex1[1,]
>
>            a          b           c          d          e          f
> 1 0.05122423 -0.5117031 -0.09961092 -0.1842569 0.03664727 -0.2285117
>
> > (tr* ex1)[1,]
>
>            a          b          c          d          e          f
> 1 0.05122423 -0.4878917 -0.1064887 -0.2445106 0.03428033 -0.2469855
>
>
> Notice that the output from tr * ex[1,] is different from (tr* ex1)[1,]
> Especially element number 4.
>
> I would naturally expect both vectors to be equivalent.
>
>
> Can anyone shed any light on my predicament??
>
> > version
>
>          _
> platform i386-pc-mingw32
> arch     i386
> os       mingw32
> system   i386, mingw32
> status
> major    1
> minor    9.1
> year     2004
> month    06
> day      21
> language R
>
>
>
> Regards
>
> Wayne Jones
>
>
>
>
>
>
> KSS Ltd
> Seventh Floor  St James's Buildings  79 Oxford Street  Manchester  M1
6SS
> England Company Registration Number 2800886
> Tel: +44 (0) 161 228 0040	Fax: +44 (0) 161 236 6305
> mailto:kssg at kssg.com		http://www.kssg.com
>
>
> The information in this Internet email is confidential and m...{{dropped}}
>
> ______________________________________________
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-- 
Arne Henningsen
Department of Agricultural Economics
University of Kiel
Olshausenstr. 40
D-24098 Kiel (Germany)
Tel: +49-431-880 4445
Fax: +49-431-880 1397
ahenningsen at agric-econ.uni-kiel.de
http://www.uni-kiel.de/agrarpol/ahenningsen/

R help - Oct 2004 - Strange Matrix Multiplication Behaviour

[R] Strange Matrix Multiplication Behaviour

[R] Strange Matrix Multiplication Behaviour

[R] Strange Matrix Multiplication Behaviour

[R] Strange Matrix Multiplication Behaviour

[R] Strange Matrix Multiplication Behaviour

[R] Strange Matrix Multiplication Behaviour

[R] Strange Matrix Multiplication Behaviour

[R] Strange Matrix Multiplication Behaviour