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# Positive Eigenvalued Matrix

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.

 matrix type OEIS counts (0,1)-matrix A003024 1, 3, 25, 543, 29281, ... (-1,0,1)-matrix A085506 1, 5, 133, 18905, ...

Weisstein's conjecture proposed that positive eigenvalued -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

with (Harary and Palmer 1973, p. 19; Robinson 1973, pp. 239-273).

Acyclic Digraph, Eigenvalue, Positive Definite Matrix, Positive Matrix, Positive Semidefinite Matrix, Weisstein's Conjecture

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## References

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 Sequences 7, 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."

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Positive Eigenvalued Matrix

## Cite this as:

Weisstein, Eric W. "Positive Eigenvalued Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PositiveEigenvaluedMatrix.html