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Proceedings of the American Mathematical Society

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Positive definite matrices and Catalan numbers

Authors: Frank Thomson Leighton and Morris Newman
Journal: Proc. Amer. Math. Soc. 79 (1980), 177-181
MSC: Primary 15A36; Secondary 05A15, 15A48
MathSciNet review: 565333
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Abstract: It is shown that the number of $ n \times n$ integral triple diagonal matrices which are unimodular, positive definite and whose sub and super diagonal elements are all one, is the Catalan number $ (_n^{2n})/(n + 1)$. More generally, it is shown that if A is a fixed integral symmetric matrix and d is a fixed positive integer, then there are only finitely many integral diagonal matrices D such that $ A + D$ is positive definite and $ \det (A + D) = d$.

References [Enhancements On Off] (What's this?)

  • [1] Morris Newman, Integral matrices, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 45. MR 0340283
  • [2] Richard Stanley, Problems and Solutions: Solutions of Elementary Problems: E2546, Amer. Math. Monthly 83 (1976), no. 10, 813–814. MR 1538211

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Keywords: Catalan number, congruence, determinant, integer matrix, positive definite matrix, triple diagonal matrix, unimodular matrix
Article copyright: © Copyright 1980 American Mathematical Society

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