Another look at a matrix of Mark Kac. (English) Zbl 0727.15010

The matrix considered here is the \((n+1)\times (n+1)\) triple diagonal matrix \(S_ n\) which has superdiagonal 1,2,...,n, subdiagonal n,n- 1,...,2,1 and zeros elsewhere. The authors give a historical survey and give new proofs of the results of M. Kac [“Probability and related topics in physical sciences” (1959; Zbl 0087.330)] and P. Rózsa [Magyar. Tud. Akad. mat. fiz. Tud. Oszt. Közleményei 7, 199-206 (1957; Zbl 0101.254)]: the characteristic values of \(S_ n\) are \(\pm n\), \(\pm (n-2),...\), where the sequence ends with \(\pm 1\) when n is odd and \(\pm 2\), 0 when n is even; letting each column eigenvector have leading element 1, if U is the matrix of column eigenvectors of \(S_ n\) corresponding to the ordering \(n,n-2,...,-n+2,n\) of the eigenvalues then \(U^ 2=2^ nI\). Finally they give proofs of two binomial identities of the first author: \[ \frac{1}{2n+1} \sum^{n}_{j=1}\binom{2n+1}{n+j+1}(2j+1)^ 2=2^{2n} \text{ and } \sum^{n}_{k=j}\binom{2n}{n+k}\frac{2k+1}{n+k+1}= \binom{2n}{n+j}. \]


15B57 Hermitian, skew-Hermitian, and related matrices
15A18 Eigenvalues, singular values, and eigenvectors
05A10 Factorials, binomial coefficients, combinatorial functions
Full Text: DOI


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