Re: characteristic polynomial

"Andreas Thom" <thom@xxxxxxxxxxxxxxxx> writes:


let p(t) be a monic polynomial with integer coefficients.

i have a question about the richness of the set of matrices that have
p(t) as characteristic polynomial.
first of all, there is a normal matrix A (normal means A*A=AA*) with
characteristic polynomial p(t). (just write the roots on the
diagonal). secondly, there is an integer matrix with characteristic
polynomial p(t). (take the companion matrix).

question 1:
is there a normal matrix with integer coefficients which has
characteristic polynomial p(t).

question 2:
if question 1 fails to be true, does there exist a positive integer k,
such that p(t)^k has the property.

(the same could be asked if p(t) has only real roots and "normal" is
replaced by "self-adjoint". that seems to be difficult as well.)

Consider the case of 2 x 2 matrices.
The normal real 2 x 2 matrices come in two families:
[ a b ]
[ b c ]
with characteristic polynomial t^2 - (a + c) t + a c - b^2, and
[ a b ]
[ -b a ]
with characteristic polynomial t^2 - 2 a t + a^2 + b^2.
If m is odd, p(t) = t^2 - m t + n can't fit the second case, while
for the first case we must have n = a (m - a) - b^2 < m^2/4. So
p(t) = t^2 - m t + n with m odd and n > m^2/4 is a counterexample for
question 1.
--
Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

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