In hoc signo vinces (Was: Revamping the numeric classes)

[Joe Fasel]

On 12-Feb-2001 William Lee Irwin III wrote:
| On Mon, Feb 12, 2001 at 02:13:38PM -0700, Joe Fasel wrote:
|> signum does make sense.  You want abs and signum to obey these laws:
|> 
|>         x == abs x * signum x
|>         abs (signum x) == (if abs x == 0 then 0 else 1)
|> 
|> Thus, having fixed an appropriate matrix norm, signum is a normalization
|> function, just as with reals and complexes.
| 
| This works fine for matrices of reals, for matrices of integers and
| polynomials over integers and the like, it breaks down quite quickly.
| It's unclear that in domains like that, the norm would be meaningful
| (in the sense of something we might want to compute) or that it would
| have a type that meshes well with a class hierarchy we might want to
| design. Matrices over Z/nZ for various n and Galois fields, and perhaps
| various other unordered algebraically incomplete rings explode this
| further still.

Fair enough.  So, the real question is not whether signum makes sense,
but whether abs does.  I guess the answer is that it does for matrix rings
over division rings.

Cheers,
--Joe

Joseph H. Fasel, Ph.D.              email: jhf@lanl.gov
Technology Modeling and Analysis    phone: +1 505 667 7158
University of California            fax:   +1 505 667 2960
Los Alamos National Laboratory      post:  TSA-7 MS F609; Los Alamos, NM 87545