NetNews Usenet Archive 1992 #19/NN_1992_19.iso/spool/sci/math/10633

Internet Message Format | 1992-08-27 | 1.7 KB

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From: [e-mail] (Bob Collins)
Newsgroups: sci.math
Subject: polynomial roots and zero divisors
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Date: 27 Aug 92 16:45:00 GMT
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  I would like to find the eigenvalues/vectors of a square
matrix.  Unfortunately, the elements of the matrix are not
members of a field.  In particular, they are "dual numbers",
defined in Yaglom's "Complex Numbers in Geometry" as a type of
complex number (a + b E), a and b being reals, and E being a
nilpotent imaginary unit; so E^2 = 0.  Dual numbers form a
commutative ring, but they are not a field since any number
of the form (0 + a E) is a zero divisor.

Question: Can I find the eigenvalues of a matrix (roots of a
characteristic polynomial) whose elements (coefficients) are
from a commutative ring, but not a field?

In the textbook "Contemporary Abstract Algebra" by J.A.Gallian
there is an example of finding all solutions to x^2 - 4 x + 3 = 0
over Z(12).  There are four solutions: x = 1, 3, 7, 9.  Gallian
says the "easiest" way to find these solutions is to try every
element!  OK for a finite group, I guess, but not for my problem.
Are there any other approaches that will work for infinite groups?

I know there is a vast literature on commutative rings out there.
Unfortunately, this isn't my field (no pun intended), so I have no
idea where to start looking.  Does anyone have any advice, or
pointers to relevant papers?


  --Bob Collins
  [e-mail]