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

**Joe Fasel
**
jhf@lanl.gov

*Mon, 12 Feb 2001 14:51:52 -0700 (MST)*

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