A127503 -id:A127503 - cp's OEIS frontend

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.

1, 2, 8, 45, 338, 3178, 34346, 396659, 4694705

Offset: 1

N. J. A. Sloane, Jul 13 2003

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.

			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.

Index entries for sequences related to binary matrices

Cf. A084552, A085658, A080858, A083029.

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

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)

cp's OEIS Frontend

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.

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