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Redheffer Matrix

Weisstein, Eric W.

A Redheffer matrix is a square (0,1)-matrix with elements a_(ij) equal to 1 if j=1 or i|j (i divides j), and 0 otherwise. For n=1, 2, ..., the first few Redheffer matrices are

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

255x255 Redheffer matrix

The Redheffer matrix of order 255 is illustrated above.

The determinant of the n×n Redheffer matrix is equal to the Mertens function M(n). For n=1, 2, ..., the first few values are therefore 1, 0, -1, -1, -2, -1, -2, -2, -2, ... (OEIS A002321).

The number of unit eigenvalues of the n×n Redheffer matrix for n>1 is equal to

a(n)=n-|_lgn_|-1

(Vaughan 1993, 1996; Trott 2004, p. 57), giving the first few values as 1, 0, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, ... (OEIS A083058).

See also

(0,1)-Matrix, Mertens Function

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References

Sloane, N. J. A. Sequences A002321/M0102 and A083058 in "The On-Line Encyclopedia of Integer Sequences."Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. https://www.mathematicaguidebooks.org/.Vaughan, R. C. "On the Eigenvalues of Redheffer's Matrix. I." In Number Theory with an Emphasis on the Markov Spectrum (Provo, UT, 1991) (Ed. A. D. Pollington and W. Moran). New York: Dekker, pp. 283-296, 1993.Vaughan, R. C. "On the Eigenvalues of Redheffer's Matrix. II." J. Austral. Math. Soc. 60, 260-273, 1996.

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Redheffer Matrix

Cite this as:

Weisstein, Eric W. "Redheffer Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RedhefferMatrix.html

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