Re: [boost] different matrix library?

Subject: Re: [boost] different matrix library?
From: DE (satan66613_at_[hidden])
Date: 2009-08-14 13:43:27

on 14.08.2009 at 21:23
 Edward Grace wrote :
> Just to stick my oar in -- there is also a subtle difference between
> a covector and a vector. This rarely seems to get a look-in when  
> people implement linear algebra stuff.  They may well be represented  
> as a tuple in both cases but they interact differently.  For example.
> covector*vector = scalar [inner product]
> vector*covector = tensor [outer product]
> Usually they are represented as columns (vectors) and rows (covectors).

> a) 1 2 3

> b) 1
>     2
>     3

> a*b = 14 (inner product)
> b*a =
>       1     2     3
>       2     4     6
>       3     6     9
> (outer product) [N.B. there are still only 6 independent components]
well if we consider a vector to be a column vector then inner product
will be

    transpose(vector)*vector    //covector*vector; actually here we
                                //get 1x1 matrix

and outer product

    vector*transpose(vector)    //vector*covector

since a vector is actually a matrix we can transpose it

> http://en.wikipedia.org/wiki/ 
> Einstein_notation#Common_operations_in_this_notation
> It would be great if these objects could be made to 'do the right
> thing', MATLAB does, mostly (it's MAT lab after all not TENS lab).
> Being able to write stuff like:
>    vector w,u,v; // They are all (column) vectors.
>    ...
>    w = levi_civita*u*v;
> and get out the conventional cross product u x v when u and v have
> length 3, as a special case of exterior products, or,
>    w = wedge(u,v); // Wedge product, function form
>    w = u ^ v;      // Wedge product operator form.
> Though I'm not sure it has the right precedence properties...
> would be pretty neat.
> I'll go away and hide now....
unfortunately i think in linalg domain (you think in tensor domain i
guess) so we may misanderstand each other
in the end i think it's possible to attach a mechanism to support
tensor style operations
by the way
>    w = u ^ v;
it's terribly wrong to overload '^' in such a way
the following is much MUCH more better solution:
>    w = wedge(u,v);

-- 
Pavel