[vox] OT: matrix operations

[Don Werve]

On 02/20/2007, at 7:57, Henry House wrote:

> Why does one never hear about a scalar addition operation? It would  
> make sense to me to define the following:
>
> [1 2]       [3 4]
> [3 4] + 2 = [5 6]

Well, you could define such an operation, but mathematically it isn't  
necessary; the above operation is equivalent to:

> [1 2]    [1 1]   [3 4]
> [3 4] + 2[1 1] = [5 6]

Defining an extra addition operation doesn't have any place in the  
axioms governing vector spaces, either, so it would just be  
'something extra'.  Assuming it's defined as my above example  
illustrates, I can't think of any way to break 'scalar addition'  
without violating the underlaying ring axioms, so it should be solid  
mathematically.

> Similarly, why is there no scalar exponentiation on vectors?

Do you mean something like the following:

   [1 2]       [1 2][1 2][1 2]
( [3 4] )^3 = [3 4][3 4][3 4]

If that's the case, this sort of operation has been around for quite  
some times, and stems from the set of NxN matrices being a ring.  If  
you're talking about non-square matrices, than exponentiation would  
be undefined, because the product:

[1 2 3][1 2 3]
[4 5 6][4 5 6]

is undefined.

Hope that helps.