## 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: $\frac{1}{2n+1} \sum^{n}_{j=1}\binom{2n+1}{n+j+1}(2j+1)^ 2=2^{2n} \text{ and } \sum^{n}_{k=j}\binom{2n}{n+k}\frac{2k+1}{n+k+1}= \binom{2n}{n+j}.$

### MSC:

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

### Citations:

Zbl 0087.330; Zbl 0101.254
Full Text:

### References:

  Cayley, A., On the determination of the value of a certain determinant, Quart. J. Math.. (Collected Mathematical Papers, Vol. 3 (1919), Cambridge U.P), 2, 120-123 (1857)  Chrystal, G., Textbook of Algebra (1919), A&C Black: A&C Black London, Part 2 · JFM 21.0073.01  Faddeev, D. K.; Sominskii, I. S., Problems in Higher Algebra (1965), Freeman: Freeman San Francisco, (transl. by J.L. Brenner) · Zbl 0125.00702  Gantmacher, F. R.; Krein, M. G., Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme (1960), Akademie-Verlag: Akademie-Verlag Berlin, 1941, 1950 · Zbl 0088.25103  Hall, H. S.; Knight, S. R., Higher Algebra (1927), Macmillan: Macmillan London  Hess, F. G., Alternative solution to the Ehrenfest problem, Amer. Math. Monthly, 61, 323-327 (1954) · Zbl 0055.37004  Kac, M., A history-dependent random sequence defined by Ulam, Adv. in Appl. Math., 10, 270-277 (1989) · Zbl 0685.60018  Kac, M., Probability and Related Topics in the Physical Science (1959), Interscience: Interscience New York  Kac, M., Random walk and the theory of Brownian motion, Amer. Math. Monthly. (Abbott, J. C., The Chauvenet Papers, Vol. 1 (1978)), 54, 253-277 (1947), together with an Appendix in which Kac refers to the papers , and also reproduces , about which he learner ned from a paper of I. Vincze .) · Zbl 0031.22604  Muir, T., A Treatise on the Theory of Determinants (1882), Macmillan: Macmillan London, (This book was published when the author was a mathematics master at Glasgow High School.) · JFM 14.0101.02  Muir, T., A Treatise on the Theory of Determinants (1960), Dover: Dover New York, rev. and enlarged by W.H. Metzler, Longmans Green, 1933  Muir, T., The Theory of Determinants in the Historical Order of Development, ((1960), Dover: Dover New York). (Contributions to the History of Determinants 1900-1920 (1930), Blackie: Blackie London), reprinted in 2 vols. · JFM 45.1242.04  Rózsa, P., Bemerkungen über die Spektralzerlegung einer stochastischen Matrix, Magyar. Tud. Akad. Mat. Fiz. Oszt. Közl.. Magyar. Tud. Akad. Mat. Fiz. Oszt. Közl., Zbl., 101, 254-206 (1963), MR 20 # 3174 · Zbl 0101.25404  Siegert, A. J.F., On the approach to statistical equilibrium, Phys. Rev., 76, 1708-1714 (1949) · Zbl 0036.14102  Schrödinger, E., Quantisierung als Eigenwertproblem (3), Ann. Phys.. (Collected Papers, Vol. 3 (1984), Österreich Akad. Wiss. Math: Österreich Akad. Wiss. Math Vienna), 80, 166-219 (1925), esp. p. 210  Sylvester, J. J., Théorèm sur les déterminants de M. Sylvester, Nouv. Ann. Math.. (Collected Mathematical Papers, Vol. 2 (1908), Cambridge U.P), 13, 28 (1854)  Vincze, I., Über das Ehrenfestsche Modell der Wärmeübertragung, Arch. Math. (Basel), 15, 394-400 (1964) · Zbl 0122.13704  Wilkinson, J. H., The Algebraic Eigenvalue Problem (1965), Oxford U.P · Zbl 0258.65037  Wilkinson, J. H., The perfidious polynomial, (Golub, G. H., MAA Stud. Math., 24 (1984), Math. Assoc. Amer: Math. Assoc. Amer Washington), 1-28 · Zbl 0601.65028  Williamson, J., The latent roots of a matrix of special type, Bull. Amer. Math. Soc., 37, 585-590 (1931) · JFM 57.0116.01  Taussky, Olga; Zassenhaus, H., On the similarity transformation between a matrix and its transpose, Pacific J. Math., 9, 893-896 (1959) · Zbl 0087.01501  Kac, M., Enigmas of Chance. An Autobiography (1985), Harper & Row: Harper & Row New York · Zbl 0604.01008  Askey, Richard; Wilson, James, A set of orthogonal polynomials that generalize the Racah coefficients or $$6-j$$ symbols, SIAM J. Math. Anal., 10, 1008-1016 (1979) · Zbl 0437.33014  Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance-Regular Graphs, (Ergeb. Math. Grenzgeb. (3), 18 (1989), Springer-Verlag: Springer-Verlag New York) · Zbl 0747.05073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.