Abstract
Let \({p_1 < p_2 < \cdots < p_n}\) be positive real numbers. It is shown that the matrix whose i, j entry is \({(p_i + p_j)^{p_i+p_j}}\) is infinitely divisible, nonsingular, and totally positive.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ando T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)
R. B. Bapat and T. E. S. Raghavan, Nonnegative Matrices and Applications, Cambridge University Press, 1997.
Bhatia R.: Infinitely divisible matrices. Amer. Math. Monthly 113, 221–235 (2006)
R. Bhatia, Positive Definite Matrices, Princeton University Press, 2007.
Bhatia R.: Min matrices and mean matrices. Math. Intelligencer 33, 22–28 (2011)
R. Bhatia and T. Jain, Inertia of the matrix \({[(p_i + p_j)^r]}\) , J. Spectral Theory, to appear.
Choi M.D.: Tricks or treats with the Hilbert matrix. Amer. Math. Monthly 90, 301–312 (1983)
S. Fallat and C. R. Johnson, Totally Nonnegative Matrices, Princeton University Press, 2011.
R. Horn and C.R. Johnson, Matrix Analysis, Second ed., Cambridge University Press, 2013.
S. Karlin, Total Positivity, Stanford University Press, 1968.
A. Pinkus, Totally Positive Matrices, Cambridge University Press, 2010.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bhatia, R., Jain, T. Positivity properties of the matrix \({\left[(i+j)^{i+j}\right]}\) . Arch. Math. 103, 279–283 (2014). https://doi.org/10.1007/s00013-014-0686-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-014-0686-5