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Positive Eigenvalued Matrix

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The numbers of positive definite n×n matrices of given types are summarized in the following table. For example, the three positive eigenvalues 2×2 (0,1)-matrices are

[1 0; 0 1],[1 0; 1 1],[1 1; 0 1],

all of which have eigenvalue 1 with degeneracy of two.

matrix typeOEIScounts
(0,1)-matrixA0030241, 3, 25, 543, 29281, ...
(-1,0,1)-matrixA0855061, 5, 133, 18905, ...

Weisstein's conjecture proposed that positive eigenvalued (0,1)-matrices were in one-to-one correspondence with labeled acyclic digraphs on n nodes, and this was subsequently proved by McKay et al. (2003, 2004). Counts of both are therefore given by the beautiful recurrence equation

a_n=sum_(k=1)^n(-1)^(k-1)(n; k)2^(k(n-k))a_(n-k)

with a_0=1 (Harary and Palmer 1973, p. 19; Robinson 1973, pp. 239-273).

See also

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 (0,1)-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 (0,1)-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

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