A positive eigenvalued matrix is a matrix whose eigenvalues are positivereal numbers.
The numbers of such matrices of given types
are summarized in the following table for , 2, .... For example, the three positive eigenvalued (0,1)-matrices are
all of which have eigenvalue 1 with multiplicity 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. https://arxiv.org/abs/math/0310423.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." J. Integer Sequences7,
Article 04.3.3, 1-5, 2004. https://cs.uwaterloo.ca/journals/JIS/VOL7/Sloane/sloane15.html.Robinson,
R. W. "Counting Labeled Acyclic Digraphs." In New
Directions in the Theory of Graphs (Ed. F. Harary). New York: Academic
Press, pp. 239-273, 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 
, 2, .... For example, the three positive eigenvalued 
(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 
vertices, and this was subsequently
proved by McKay et al. (2003, 2004). The correspondence maps a labeled acyclic digraph with adjacency
matrix 
to 
,
where 
is the identity matrix. All eigenvalues
of the resulting matrix are 1. Counts of both are therefore given by the recurrence
equation
,
with 
(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.