A038379 -id:A038379 - cp's OEIS frontend

A083029 Triangle read by rows: T(n,k), n >=1, 0 <= k <= C(n,k), = number of n X n symmetric positive semi-definite matrices with 2's on the main diagonal and 1's and 0's elsewhere and with k 1's above the diagonal.

1, 1, 1, 1, 3, 3, 1, 1, 6, 15, 20, 15, 6, 1, 1, 10, 45, 120, 210, 192, 200, 90, 45, 10, 1, 1, 15, 105, 455, 1365, 2517, 3805, 3975, 3690, 2910, 1548, 975, 255, 105, 15, 1, 1, 21, 210, 1330, 5985, 18207, 39557, 71235, 95130, 115115, 110670, 104265, 72520, 56070, 32445, 15862, 7434, 2730, 665, 210, 21, 1, 1, 28, 378, 3276, 20475, 91392, 288596, 692576, 1374597, 2161180, 3247622, 3740016, 4422915, 4117512, 3886200, 3044048, 2579780, 1591296, 1111768, 628600, 323148, 148184, 65576, 20160, 7105, 1540, 378, 28, 1

Offset: 1

N. J. A. Sloane, Jul 13 2003

			1;
1,1;
1,3,3,1;
1,6,15,20,15,6,1;
...

Index entries for sequences related to binary matrices

Rows sums give A085658.
A038379(n) = Sum_{k=0..C(n,2)} 2^k * T(n,k).
A084546 is an upper bound.

Rows n=6..8 added by Max Alekseyev, Jun 05 2024

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

Original entry on oeis.org

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

A085658 Number of n X n symmetric positive semi-definite matrices with 2's on the main diagonal and 1's and 0's elsewhere.

Original entry on oeis.org

1, 2, 8, 64, 924, 21737, 749684, 33568376

Offset: 1

N. J. A. Sloane, Jul 12 2003

Of course the total number of symmetric matrices of this type (not necessarily positive semi-definite) is 2^C(n,2).

This gives the number of different values of M + M' where M runs through the matrices counted in A038379. - Max Alekseyev, Nov 11 2006

			The matrix
2 0 0 0 1
0 2 0 1 1
0 0 2 1 1
0 1 1 2 0
1 1 1 0 2
is one of the 100 5 X 5 matrices which are not positive semi-definite.
Its eigenvalues are approximately [2., -0.135779205069857, 4.135779205069857, 1.337846553138044, 2.662153446861956]

Index entries for sequences related to binary matrices

Cf. A085658, A080858, A083029.

3 more terms from Max Alekseyev, Nov 08 2006

A127502 Number of n X n positive definite matrices with 1's on the main diagonal and -1's and 0's elsewhere.

Original entry on oeis.org

1, 3, 19, 201, 3001, 55291, 1115003, 21837649, 373215601, 8282131891

Offset: 1

Max Alekseyev, Jan 16 2007

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 A084552.

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

Cf. A085656, A085657, A085658, A086215, A038379, A080858, A083029, A127503.

{ 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,[ -1,0]), 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)

A127503 Number of n X n matrices A with 1's on the main diagonal and -1's and 0's elsewhere such that A + A' has only 0's and -1's off the main diagonal and is positive semi-definite.

Original entry on oeis.org

1, 3, 27, 281, 3945, 70635, 1437555, 30357425, 628337745

Offset: 1

Max Alekseyev, Jan 16 2007

For number of different values of A + A' see A084553.

Cf. A085657, A085658, A080858, A083029, A038379, A127502.

cp's OEIS Frontend

A083029 Triangle read by rows: T(n,k), n >=1, 0 <= k <= C(n,k), = number of n X n symmetric positive semi-definite matrices with 2's on the main diagonal and 1's and 0's elsewhere and with k 1's above the diagonal.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A085658 Number of n X n symmetric positive semi-definite matrices with 2's on the main diagonal and 1's and 0's elsewhere.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A127502 Number of n X n positive definite matrices with 1's on the main diagonal and -1's and 0's elsewhere.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A127503 Number of n X n matrices A with 1's on the main diagonal and -1's and 0's elsewhere such that A + A' has only 0's and -1's off the main diagonal and is positive semi-definite.

Views

Author

Keywords

Comments

Crossrefs