# [seqfan] Re: Number of n x n {-1, 1}-matrices which have only eigenvalues with strictly negative real part.

Alex Meiburg timeroot.alex at gmail.com
Mon Aug 14 16:20:42 CEST 2023

```If you negate each element, you negate each eigenvalue, so then you're
looking for strictly positive eigenvalues. That seems a bit more natural to
count.

If the matrix were symmetric, then there's the rich theory of PSD matrices.
Sylvester's criterion, for instance, would be a powerful way to search
quickly by computer. But the theory falls apart quickly when the matrices
aren't symmetric. And what attention there is, is about matrices with xMx >
0, not positive real parts of eigenvalues (two conditions which are
equivalent only if the matrix is symmetric).

On Mon, Aug 14, 2023, 08:49 <israel at math.ubc.ca> wrote:

> Since the trace is the sum of the eigenvalues, it's certainly true that
> the entries of -1 must be the majority on the main diagonal.
>
> Cheers,
> Robert
>
> On Aug 14 2023, Thomas Scheuerle via SeqFan wrote:
>
> >Hello Seqfan list,
> >
> >I have planned to prepare a sequence
> >which enumerates all matrices where all entries are -1 or 1
> >and the eigenvalues have real parts < 0.
> >
> >Two of the 20  3 x 3 matrices would be:
> >    -1     1     1
> >    -1    -1     1
> >    -1     1    -1
> >
> >    -1     1     1
> >     1    -1     1
> >    -1    -1    -1
> >            ...
> >By naively counting I got for a(1) to a(5) already this values:
> >1, 2, 20, 640, 97824, ...
> >
> > I noticed that there are likely many algorithms which are faster and
> more
> > sophisticated. Also I expected that this problem was already researched
> > somewhere O
> >
> > As my knowledge is very limited I propose this topic here to allow some
> > brainstorming of the crowd.
> >
> > The first improvement could be based on the main diagonal and the trace
> > of the matrices. It appears that negative entries on the main diagonal
> > must be at least in the majority. n = 5 is the least n where not all
> > entries on the main diagonal are -1.
> >
> >Also I lack the sufficient knowledge to decide which symmetries are valid.
> >
> > Another idea would be some heuristics to exclude determinant = 0 somehow
> > fast ?
> >
> >But frankly I have not any good concepts yet.
> >
> >Thank you very much
> >
> >Thomas Scheuerle
> >
> >--
> >Seqfan Mailing list - http://list.seqfan.eu/
> >
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
```