Positive definite matrices and Catalan numbers
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- by Frank Thomson Leighton and Morris Newman PDF
- Proc. Amer. Math. Soc. 79 (1980), 177-181 Request permission
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
- Morris Newman, Integral matrices, Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972. MR 0340283
- Richard Stanley, Problems and Solutions: Solutions of Elementary Problems: E2546, Amer. Math. Monthly 83 (1976), no. 10, 813–814. MR 1538211
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 79 (1980), 177-181
- MSC: Primary 15A36; Secondary 05A15, 15A48
- DOI: https://doi.org/10.1090/S0002-9939-1980-0565333-4
- MathSciNet review: 565333