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

israel at math.ubc.ca israel at math.ubc.ca
Mon Aug 14 15:49:27 CEST 2023

```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/
>
>
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