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

[Max A.]

On 1/14/07, Artur <grafix at csl.pl> wrote:
> Let zeroes of characteristic polynomial of matrix A will be respectively> (a1,a2,...an) and matrix B will be (b1,b2,...,bn)> now trace A is a1+a2+...an  trace B is b1+b2+...bn> det A =a1*a2*..*an     det B=b1*b2...*bn>> and now your proof with number 2n! will be good if doesn't existed case> if (a1*a2*..*an=b1*b2...*bn) and (a1+a2+...an)=(b1+b2+...bn)> don't occured such case that> (a1*a2+a1*a3+...+a1*an+a2*a3+...)<>(b1*b2+b1*b3+...+b1*bn+b2*b3+...)
That cannot happen. According to the statement 2),if (a1*a2*..*an=b1*b2...*bn) and (a1+a2+...an)=(b1+b2+...bn) then B=PAP^{-1} or> B=(PAP^{-1})^T where P is some permutation matrix, which in turn implies that charpoly(A)=charpoly(B) and, in particular:(a1*a2+a1*a3+...+a1*an+a2*a3+...) = (b1*b2+b1*b3+...+b1*bn+b2*b3+...)
Max
> Dnia 14-01-2007 o 01:13:18 Max A. <maxale at gmail.com> napisał(a):>> > The following statements (suggested by lofar in russian forum> > lib.mexmat.ru/forum/) are true:> >> > Let n x n matrices A and B contain the same n^2 variables as elements,> > then> > 1) If |det(A)|=|det(B)| then B=PAQ or B=(PAQ)^T where P and Q are some> > permutation matrices.> > 2) If |det(A)|=|det(B)| and trace(A)=trace(B) then B=PAP^{-1} or> > B=(PAP^{-1})^T where P is some permutation matrix.> >> > Now it is easy to give an answer to your original question:> >> >> How many different matrices n x n with n^2 different elements occured> >> which have that same characteristic polynomial?> >> > Note that if charpoly(A)=charpoly(B) then det(A)=det(B) and> > trace(A)=trace(B), implying that B=PAP^{-1} or B=(PAP^{-1})^T for some> > permutation matrix P. Therefore, the answer to your question is 2*n!> > (where n! stands for the number of different permutation matrices and> > 2 accounts for a possible transposition of the matrix).> >> > Max> >> > On 1/8/07, Artur <grafix at csl.pl> wrote:> >> Dear Seqfans,> >> I'm asking: How many different matrices n x n with n^2 different> >> elements> >> occured which have that same characteristic polynomial> >> We have to count all permutations n^2 elements in n x n matrix and count> >> only these permutations> >> which don't changed starting polynomial> >> for 2 x 2 case> >> we have 4 matrices X^2-(a+d)X+ad-bc> >> a b   a c   d c   d b> >> c d   b d   b a   c a> >>> >> BEST WISHES> >> ARTUR> >>> >>>>>