A306791 -id:A306791 - cp's OEIS frontend

A306794 Number of distinct real eigenvalues of n X n matrices with elements {-1, 1}.

2, 5, 15, 119, 2297, 145213

Offset: 1

Steven E. Thornton, Mar 10 2019

S. E. Thornton, Properties of the Bohemian family of n x n matrices with population {-1, 1}, Characteristic Polynomial Database.

Number of distinct eigenvalues is in A306791.
Number of characteristic polynomials is in A272662.
Number of minimal polynomials is in A306782.

A306819 Number of nilpotent n X n matrices with elements {-1, 1}.

Original entry on oeis.org

0, 4, 0, 2064, 0, 100172800

Offset: 1

Steven E. Thornton, Mar 11 2019

S. E. Thornton, Properties of the Bohemian family of n x n matrices with population {-1, 1}, Characteristic Polynomial Database.

Number of characteristic polynomials is in A272662.

Number of distinct eigenvalues is in A306791.

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

A306794 Number of distinct real eigenvalues of n X n matrices with elements {-1, 1}.

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A306819 Number of nilpotent n X n matrices with elements {-1, 1}.

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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).

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