A085506 -id:A085506 - cp's OEIS frontend

A085656 Number of positive-definite real {0,1} n X n matrices.

1, 3, 27, 681, 43369, 6184475, 1688686483, 665444089745

Offset: 1

N. J. A. Sloane, Jul 12 2003

A real matrix M is positive-definite if x M x' > 0 for all nonzero real vectors x. Equivalently, all eigenvalues of M + M' are positive.

M need not be symmetric. For the number of different values of M + M' see A085657. - Max Alekseyev, Dec 13 2005

			For n = 2 the three matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}}.

Eric Weisstein's World of Mathematics, (0,1)-Matrix
Eric Weisstein's World of Mathematica, Positive Definite Matrix
Index entries for sequences related to binary matrices

Cf. A055165, which counts nonsingular {0, 1} matrices and A085506, which counts {-1, 0, 1} matrices with positive eigenvalues.

Cf. A085657, A085658, A086215, A038379 (positive semi-definite matrices), A080858, A083029.

Table[Count[Tuples[{0, 1}, {n, n}], ?PositiveDefiniteMatrixQ], {n, 4}] (* _Eric W. Weisstein, Jan 03 2021 *)

{ a(n) = M=matrix(n,n,i,j,2*(i==j)); r=0; b(1); r } { b(k) = local(t); if(k>n, t=0; for(i=1,n, for(j=1,i-1, if(M[i,j]==1,t++); )); r+=2^t; return; ); forvec(x=vector(k-1,i,[0,1]), for(i=1,k-1,M[k,i]=M[i,k]=x[i]); if( matdet(vecextract(M,2^k-1,2^k-1),1)>0, b(k+1) ) ) } (Alekseyev)

More terms from Max Alekseyev, Dec 13 2005

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

A085656 Number of positive-definite real {0,1} n X n matrices.

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