cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

N. J. A. Sloane, Jul 12 2003

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

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

Extensions

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

Views

Author

N. J. A. Sloane, Jul 12 2003

Keywords

Comments

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

Examples

			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]

Links

Crossrefs

Cf. A085658, A080858, A083029.

Extensions

3 more terms from Max Alekseyev, Nov 08 2006

A085657 Number of n X n symmetric positive definite matrices with 2's on the main diagonal and 1's and 0's elsewhere.

Original entry on oeis.org

1, 2, 8, 61, 819, 17417, 506609, 15582436
Offset: 1

Views

Author

N. J. A. Sloane, Jul 12 2003

Keywords

Comments

Of course the total number of symmetric matrices of this type (not necessarily positive definite) is 2^C(n,2).
This gives the number of different values of A + A' where A runs through the matrices counted in A085656. - Max Alekseyev, Dec 13 2005

Examples

			The singular matrix
2 0 1 1
0 2 1 1
1 1 2 0
1 1 0 2
is one of the three 4 X 4 matrices which are not positive definite.

Links

Crossrefs

Cf. A085659, A080858, A083029.
Cf. A085656, A114601.

Programs

  • PARI
    { a(n) = M=matrix(n,n,i,j,2*(i==j)); r=0; b(1); r } { b(k) = if(k>n, r++; 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) ) ) } (Max Alekseyev)

Extensions

More terms from Max Alekseyev, Dec 13 2005

A038379 Number of real {0,1} n X n matrices A such that M = A + A' has 2's on the main diagonal, 0's and 1's elsewhere and is positive semi-definite.

Original entry on oeis.org

1, 3, 27, 729, 52649, 9058475, 3383769523, 2520512534065
Offset: 1

Views

Author

N. J. A. Sloane, Jul 13 2003

Keywords

Comments

Necessarily A has all 1's on the main diagonal.
A real matrix M is positive semi-definite if its eigenvalues are >= 0.
For n <= 4, a(n) equals the upper bound 3^C(n,2).
For the number of different values of symmetric parts A + A', see A085658. - Max Alekseyev, Nov 11 2006

Links

Crossrefs

Cf. A055165, which counts nonsingular {0, 1} matrices, A003024, which counts {0, 1} matrices with positive eigenvalues, A085656 (positive definite matrices).
Cf. A085657, A085658, A080858, A083029.

Formula

a(n) = Sum_{k=0..C(n,2)} 2^k * A083029(n,k).

Extensions

Definition corrected Nov 10 2006
a(6)-a(8) from Max Alekseyev, Nov 11 2006
Edited by Max Alekseyev, Jun 05 2024

A084552 Number of n X n symmetric positive definite matrices with 2's on the main diagonal and -1's and 0's elsewhere.

Original entry on oeis.org

1, 2, 7, 38, 286, 2686, 28512, 312572, 3337588, 40963216
Offset: 1

Views

Author

N. J. A. Sloane, Jul 13 2003

Keywords

Comments

Of course the total number of symmetric matrices of this type (not necessarily positive definite) is 2^C(n,2).
This gives the number of different values of M + M' where M runs through the matrices counted in A127502.

Examples

			The singular matrix
2 -1 -1
-1 2 -1
-1 -1 2
is the only 3 X 3 matrix of this type which is not positive definite.

Links

Crossrefs

Cf. A084553, A085659, A080858, A083029.

Extensions

a(6)-a(10) from Max Alekseyev, Jan 16 2006

A084553 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, 45, 338, 3178, 34346, 396659, 4694705
Offset: 1

Views

Author

N. J. A. Sloane, Jul 13 2003

Keywords

Comments

Of course the total number of symmetric matrices of this type (not necessarily positive definite) is 2^C(n,2).
This gives the number of different values of A + A' where A runs through the matrices counted in A127503.

Examples

			The 5 X 5 matrix
2 -1 -1 -1 -1
-1 2 -1 -1 -1
-1 -1 2 -1 -1
-1 -1 -1 2 -1
-1 -1 -1 -1 2
with eigenvalues -2, 3, 3, 3, 3 is an example of one which is not positive semi-definite.

Links

Crossrefs

Cf. A084552, A085658, A080858, A083029.

Extensions

a(6)-a(9) from Max Alekseyev, Jan 16 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

Views

Author

Max Alekseyev, Jan 16 2007

Keywords

Comments

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.

Examples

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

Crossrefs

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

Programs

  • PARI
    { 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

Views

Author

Max Alekseyev, Jan 16 2007

Keywords

Comments

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

Crossrefs

Cf. A085657, A085658, A080858, A083029, A038379, A127502.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.