Re: How many normal matrices are not symmetric?

On Jan 27, 4:54 pm, Robert Israel
<isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
Gerry Myerson <ge...@xxxxxxxxxxxxxxxxxxxxxxxxx> writes:
In article
<942a6e70-a178-41e1-97ff-aed56836a...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Yaroslav Bulatov <yarosla...@xxxxxxxxx> wrote:

Suppose I look at nxn normal matrices with integer entries in -N..N
What fraction of such matrices are not symmetric if I let N go to
infinity?
What might be a practical way of estimating this limit?

Well, here's what I might do if I wanted to work this out.

First I'd go to the On-Line Encyclopedia of Integer Sequences
to search for "normal matrices". Unfortunately, that doesn't
turn up anything immediately useful - sequence A055547 is
the number of normal n x n matrices with 0, 1 entries,
A055548 is same with 1, -1 entries. Still might be worth a look.

Then I'd restrict to n = 2 and work out the number of normal
2 x 2 matrices with integer entries between -N and N for N = 1, 2, 3, ...
until I a) saw a pattern, or b) got tired, and I'd check it with
the On-Line Encyclopedia again, while also comparing it to the number
of symmetric ones. Maybe the 2 x 2 case is easy enough to solve
explicitly, that is, to find a formula that gives all of them in such
a way as to make it easy to count them. Worth a try - and if you
can't do the 2 x 2 case, what hope of doing the general case?

The 2 x 2 case is indeed easy, but perhaps too special. The 2 x 2 real matrix

[ a b ]
[ c d ]

is normal iff
(1) b = c
or
(2) b = -c and a = d.

With integer entries in [-N, N], there are (2N+1)^3 possibilities for (1)
and (2N+1)^2 for (2), with 2N+1 in the intersection, for a total of
(2N+1)^3 + (2N+1)^2 - (2N+1). The fraction that are not symmetric is thus
2N/((2N+1)^2 + 2N), which goes to 0 as N -> infinity.

For the 3 x 3 case, I tried Maple's Groebner basis commands
on the equations that make a matrix normal.
There were 34 different families of solutions, but only one
has as many as 6 free parameters, namely the symmetric matrices. The
largest class containing non-symmetric matrices
was
[ a b c ]
[ -b a d ]
[ -c -d a ]
with only 4 free parameters.
Thus there are (2N+1)^6 symmetric 3 x 3 matrices with entries in [-N,
N], but each of the non-symmetric classes of normal matrices have at
most O(N^4). So if this is right, the fraction of non-symmetric
matrices is O(N^(-2)).

--
Robert Israel isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada


.

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