The numbers of positive definite matrices of given types are summarized in the following
table. For example, the three positive eigenvalues (0,1)-matrices are
all of which have eigenvalue 1 with degeneracy of
two.
Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, 1973.McKay, B. D.;
Oggier, F. E.; Royle, G. F.; Sloane, N. J. A.; Wanless, I. M.;
and Wilf, H. "Acyclic Digraphs and Eigenvalues of -Matrices." 28 Oct 2003. http://arxiv.org/abs/math/0310423.McKay,
B. D.; Royle, G. F.; Wanless, I. M.; Oggier, F. E.; Sloane, N. J. A.;
and Wilf, H. "Acyclic Digraphs and Eigenvalues of -Matrices." J. Integer Sequences7,
Article 04.3.3, 1-5, 2004. http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Sloane/sloane15.html.Robinson,
R. W. "Counting Labeled Acyclic Digraphs." In New
Directions in Graph Theory (Ed. F. Harary). New York: Academic Press,
1973.Sloane, N. J. A. Sequences A003024/M3113
and A085506 in "The On-Line Encyclopedia
of Integer Sequences."
matrices of given types are summarized in the following
table. For example, the three positive eigenvalues 
(0,1)-matrices are
![[1 0; 0 1],[1 0; 1 1],[1 1; 0 1],](/images/equations/PositiveEigenvaluedMatrix/NumberedEquation1.svg)
-matrices
were in one-to-one correspondence with
labeled acyclic digraphs on 
nodes, and this was subsequently proved by McKay et al.
(2003, 2004). Counts of both are therefore given by the beautiful recurrence
equation
(Harary and Palmer 1973, p. 19; Robinson 1973, pp. 239-273).
-Matrices." 28 Oct 2003. 
-Matrices." J. Integer Sequences 7,
Article 04.3.3, 1-5, 2004.