OFFSET
1,2
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
LINKS
Eric Weisstein's World of Mathematics, (0,1)-Matrix
Eric Weisstein's World of Mathematica, Positive Definite Matrix
EXAMPLE
For n = 2 the three matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}}.
MATHEMATICA
Table[Count[Tuples[{0, 1}, {n, n}], _?PositiveDefiniteMatrixQ], {n, 4}] (* Eric W. Weisstein, Jan 03 2021 *)
PROG
(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)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, Jul 12 2003
EXTENSIONS
More terms from Max Alekseyev, Dec 13 2005
STATUS
approved