A271588 -id:A271588 - cp's OEIS frontend

A272658 Number of distinct characteristic polynomials of n X n matrices with elements {-1, 0, +1}.

Original entry on oeis.org

1, 3, 16, 209, 8739, 1839102

Offset: 0

N. J. A. Sloane, May 15 2016

Robert M. Corless, Bohemian Eigenvalues, Talk Presented at Computational Discovery in Mathematics (ACMES 2), University of Western Ontario, May 12 2016. (Talk based on joint work with Steven E. Thornton, Sonia Gupta, Jonathan Brino-Tarasoff, Venkat Balasubramanian.)

Eunice Y. S. Chan, Algebraic Companions and Linearizations, The University of Western Ontario (Canada, 2019) Electronic Thesis and Dissertation Repository. 6414.
Eunice Y. S. Chan and Robert Corless, A new kind of companion matrix, Electronic Journal of Linear Algebra, Volume 32, Article 25, 2017, see p. 335.
Robert M. Corless et al., Bohemian Eigenvalues.
Robert Corless and Steven Thornton, The Bohemian Eigenvalue Project, 2017 poster.

Six classes of matrices mentioned in Rob Corless's talk: this sequence, A272659, A272660, A272661, A272662, A272663.

Other properties of this class of matrices: A271570, A271587, A271588. - Steven E. Thornton, Jul 13 2016

a[n_] := a[n] = Module[{m, cPolys}, m = Tuples[Tuples[{-1, 0, 1}, n], n]; cPolys = CharacteristicPolynomial[#, x] & /@ m; Length[DeleteDuplicates[cPolys]]]; Table[a[i], {i, 1, 3}] (* Robert P. P. McKone, Sep 16 2023 *)

from itertools import product
from sympy import Matrix
def A272658(n): return len({tuple(Matrix(n,n,p).charpoly().as_list()) for p in product((-1,0,1),repeat=n**2)}) if n else 1 # Chai Wah Wu, Sep 30 2023

a(n) <= 3^(n^2). - Robert P. P. McKone, Sep 16 2023

a(4) found by Daniel Lichtblau, May 13 2016
a(5) found by Daniel Lichtblau and Steven E. Thornton, May 19 2016
a(0)=1 prepended by Alois P. Heinz, Sep 28 2023

A364886 Number of n X n (-1, 1)-matrices which have only eigenvalues with strictly negative real part (which implies that the matrix has all nonzero eigenvalues).

Original entry on oeis.org

1, 2, 20, 640, 97824, 47545088

Offset: 1

Thomas Scheuerle, Aug 12 2023

As this problem is symmetric with sign we can get the same numbers for strictly positive real parts.
All values for n > 1 are even, because a transposed matrix has the same spectrum of eigenvalues.
Matrices with determinant 0 are not counted.
Let M be such a matrix then the limit of ||exp(t*M)*y|| if t goes to infinity will be zero.
n = 5 is the first case where not all entries on the main diagonal are -1. 93984 matrices with 5 times -1 on the main diagonal and 5*768 with 4 times -1 on the main diagonal have only eigenvalues with strictly negative real part.
In the case n = 6, 43586048 matrices with 6 times -1 on the main diagonal, 6*656000 matrices with 5 times -1 on the main diagonal and 15*1536  matrices with 5 times -1 on the main diagonal have only eigenvalues with strictly negative real part.

			For n = 2 the matrices are:
.
    -1,  1
    -1, -1
.
    -1, -1
     1, -1.

Index entries for sequences related to binary matrices

Cf. A056990.
Cf. A083058, A085506, A087488, A098148, A086510, A207259.
Cf. A219736, A346210, A271570, A271588, A296605, A306002.
Cf. A306791, A306792, A306793, A306794, A306795, A326928.
Cf. A346209.

cp's OEIS Frontend

A272658 Number of distinct characteristic polynomials of n X n matrices with elements {-1, 0, +1}.

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