Number of matrices n x n with n^2 different elements which have that same characteristic polynomial

[Brendan McKay]

THEOREM.  Let A and B be n*n matrices of the same n^2 distinct
indeterminants, each appearing once. Suppose trace(A^i)=trace(B^i)
for i=1,3,4 (don't need i=2). Then for some permutation matrix P,
either PA=BP or PA^T=BP.

(Note that these conditions are weaker than having the same
characteristic polynomial.)

PROOF.  We try to reconstruct A from knowing trace(A^i) for i=1,3,4.
The value of trace(A^1) tells us which indeterminants are on the
diagonal, say x[1],x[2],...,x[n]. Permute rows and columns
equally so that x[i] is in the (i,i) position (this uses up the
possibilities for P).

Now look at trace(A^4).  If there is a term x[i]*x[j]*v*w where
i <> j and v,w are non-diagonal indeterminants, then v and w must
be in positions (i,j) and (j,i) but so far we can't tell which is
which. All the non-diagonal indeterminants can be paired up in this
fashion.

The pair for positions (1,2) and (2,1) can be inserted in two ways;
choose one (this uses up the choice of PA=BP or PA^T=BP).

Finally, look at trace(A^3).  The terms like a*b*c for non-diagonal
indeterminants a,b,c must have the form A[i,j]*A[j,k]*A[k,i] for
distinct i,j,k.  Starting with i=1,j=2, this uniquely determines
for each (i,j),(j,i) pair which indeterminate goes in (i,j) and
which goes in (j,i). EOP

PROBLEM.  Which sets of powers suffice in place of {1,3,4}?

Cheers, Brendan.



Another possible solution would be to use a forum on the web that uses
discussion group software. One would log into the forum to see the messages,
the messages could be grouped into threads and the replies to a thread 
could be
organized in a tree with the titles visible, you could click on a reply to 
expand the whole tree to see the whole thread.
There is some really good software available.
I think it is easier to navigate, if a thread doesn't interest you, you 
only see the
one title instead of multiple email titles. Even serious emails may simply 
be not
interesting (to everyone), Seeing all the replies to a subject is cumbersome.
Also many of the replies usually have the same title, in a forum that can 
vary.
The information in a thread is also more organized than back-and-forth emails
that contain varying portions of the thread, it's like having a book in bit 
and pieces.
Case in point - someone may have already suggested this for all I know!
It's hard to read all the replies.
Another thing, I for one am a bit obsessive, so I end up spending too much time
trying to understand the subject of an email thread. I think that is less 
likely
with a forum.

Gerald

At 12:38 PM 1/11/2007, Jon Awbrey wrote:
>o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>
>one solution might be to subscribe the seqfan list
>to a newsreader service like gmane:
>
>http://gmane.org/
>http://news.gmane.org/
>
>instead of your inbox filling up all day long,
>you only go to your newsreader window when you
>have time, download the headers, and read the
>ones you want.
>
>many discussion groups maintain this in parallel to the email format,
>as you can disable delivery on the list but still respond to mail
>from the newsreader window.
>
>ja
>
>o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>inquiry e-lab: http://stderr.org/pipermail/inquiry/
>zhongwen wp: http://zh.wikipedia.org/wiki/User:Jon_Awbrey
>wikinfo: http://wikinfo.org/wiki.php?title=User:Jon_Awbrey
>http://www.getwiki.net/wiki.php?title=User_talk:Jon_Awbrey
>wp review: http://wikipediareview.com/index.php?showuser=398
>o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o