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).

1, 2, 20, 640, 97824, 47545088

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..6.

Index entries for sequences related to binary matrices

Example

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

Crossrefs

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.

Sequence in context: A054941 A012495 A168480 * A198761 A171799 A251183

Adjacent sequences: A364883 A364884 A364885 * A364887 A364888 A364889

Keyword

nonn,more

Author

Thomas Scheuerle, Aug 12 2023

Status

approved