A364886 - OEIS

%I #26 Oct 18 2023 09:21:51

%S 1,2,20,640,97824,47545088

%N 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).

%C As this problem is symmetric with sign we can get the same numbers for strictly positive real parts.

%C All values for n > 1 are even, because a transposed matrix has the same spectrum of eigenvalues.

%C Matrices with determinant 0 are not counted.

%C Let M be such a matrix then the limit of ||exp(t*M)*y|| if t goes to infinity will be zero.

%C 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.

%C 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.

%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>

%e For n = 2 the matrices are:

%e .

%e     -1,  1

%e     -1, -1

%e .

%e     -1, -1

%e      1, -1.

%Y Cf. A056990.

%Y Cf. A083058, A085506, A087488, A098148, A086510, A207259.

%Y Cf. A219736, A346210, A271570, A271588, A296605, A306002.

%Y Cf. A306791, A306792, A306793, A306794, A306795, A326928.

%Y Cf. A346209.

%K nonn,more

%O 1,2

%A _Thomas Scheuerle_, Aug 12 2023