Prev: topology
Next: Universe itself is the one and only magnetic monopole Chapt 14 #182; ATOM TOTALITY
From: Kaba on 20 Jun 2010 18:29
Stephen Montgomery-Smith wrote:
> Kaba wrote:
> > Axel Vogt wrote:
> >> Kaba wrote:
> >>> Hi,
> >>>
> >>> I'd like construct matrix algebra as an algebra over a field. How would
> >>> you approach this?
> >>
> >> multiplication = composition of morphisms ?
> >
> > Yes, but what confuses me is that there are all linear transformations
> > from R^n from R^m, for all n, m in N, where N is natural numbers. And
> > (R^n -> R^m) o (R^m -> R^p) gives (R^n -> R^p) etc. I would need some
> > common object to represent them all to keep composition closed.
> > R^{N x N} comes to mind.
>
> How about using finite rank bounded linear operators on a Hilbert space
> as the common object?

I think that is more or less what I meant by the R^{N x N} (assuming
there are only finitely many non-zero entries).

Here's one try:

Let M = {A in R^{m x n} : m, n in N} union {E}, where the elements of
R^{m x n} are taken as just arrays of numbers with no additional
structure, and E is the empty set representing a special empty matrix E.
Give a vector-space structure to M by pointwise addition and
multiplication by a scalar (real number), when the extents match.
Otherwise, define the result as E.

Define * : M x M -> M: o(A, B), as the familiar matrix multiplication of
A and B, when the extents are consistent, and E otherwise. This
operation is bilinear.

Then C = (M, *) is an algebra over a field, called the matrix algebra
(over the reals).

--
http://kaba.hilvi.org
From: Kaba on 20 Jun 2010 18:42
Kaba wrote:
> Then C = (M, *) is an algebra over a field, called the matrix algebra
> (over the reals).

But damn... I failed to capture the identity element so that I could say
it is unital. So not good enough.

--
http://kaba.hilvi.org
From: Stephen Montgomery-Smith on 20 Jun 2010 18:51
Kaba wrote:
> Kaba wrote:
>> Then C = (M, *) is an algebra over a field, called the matrix algebra
>> (over the reals).
>
> But damn... I failed to capture the identity element so that I could say
> it is unital. So not good enough.
>


To do that, use the matrices that are the sum of a multiple of the
identity and a finite rank matrix. The identity HAS to be an "infinite
matrix."

First  |  Prev  | 
Pages: 1 2
Prev: topology
Next: Universe itself is the one and only magnetic monopole Chapt 14 #182; ATOM TOTALITY