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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


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$.

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PII: S 0002-9939(1980)0565333-4
Keywords: Catalan number, congruence, determinant, integer matrix, positive definite matrix, triple diagonal matrix, unimodular matrix
Article copyright: © Copyright 1980 American Mathematical Society