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Positive definite functions of noncommuting contractions, Hua-Bellman matrices, and a new distance metric

Primarily human-written textHuman understanding: all partsmath.FA — Functional Analysismath.OA — Operator Algebrasmath.MG — Metric Geometry

Contributed by Suvrit Sra ↗

Version 1 / Oct 29, 2026/ arXiv: 2112.00056v2 / CC BY 4.0

Abstract

We study positive definiteness of Hua-Bellman matrices (i.e., matrices of the form [det⁡(I−Ai∗Aj)−α]ij[\det(I-A_i^*A_j)^{-α}]_{ij}, where AiA_i and AjA_j are strict contractions and α∈Rα\in\mathbb{R}). We start by revisiting a 1959 work of Bellman (Representation theorems and inequalities for Hermitian matrices; Duke Mathematical J., 26(3), 1959) that studies positive definiteness of Hua-Bellman matrices and claims a strengthening of Hua's representation-theoretic results (Inequalities involving determinants; Acta Mathematica Sinica, 5 (1955)). We uncover a critical error in Bellman's proof that has surprisingly escaped notice to date, and we show by an explicit example that his claim itself is false. We then provide conditions under which det⁡(I−A∗B)−α\det(I-A^*B)^{-α} is a positive definite function. For complex contractions our condition is the known sharp one, with the integer exponents below Hua's range reached by an elementary argument. For real symmetric contractions we recover the part of Bellman's claims that survives. Building on our result, we introduce a new Poincaré-like distance metric on noncommuting strict contractions, and yet another `log-det' based distance in the appendix.

Provenance statement

Purely human effort: the mathematics and the text are the author's own, a revision of arXiv:2112.00056v1 (2021). AI usage was limited to minor proofreading.

References

  1. T. Ando. H ua-- M arcus inequalities. Linear and Multilinear Algebra , 8 0 (4): 0 347--352, 1980. 10.1080/03081088008817340.DOI
  2. T. Ando. Positivity of operator-matrices of H ua-type. Banach Journal of Mathematical Analysis , 2 0 (2): 0 1--8, 2008. 10.15352/bjma/1240336286.DOI
  3. Richard Bellman. Representation theorems and inequalities for H ermitian matrices. Duke Mathematical Journal , 26 0 (3): 0 485--490, 1959. 10.1215/s0012-7094-59-02645-6.DOI
  4. Alexander Belton, Dominique Guillot, Apoorva Khare, and Mihai Putinar. Preservers of totally positive kernels and P ólya frequency functions. Mathematics Research Reports , 3: 0 35--56, 2022. 10.5802/mrr.12.DOI
  5. Christian Berg, Jens Peter Reus Christensen, and Paul Ressel. Harmonic analysis on semigroups: theory of positive definite and related functions , volume 100 of Graduate Texts in Mathematics . Springer, New York, 1984. 10.1007/978-1-4612-1128-0.DOI
  6. Rajendra Bhatia. Matrix Analysis . Graduate Texts in Mathematics 169. Springer, New York, 1997. 10.1007/978-1-4612-0653-8.DOI
  7. Florent Bouchard, J é r ô me Malick, and Marco Congedo. Riemannian optimization and approximate joint diagonalization for blind source separation. IEEE Transactions on Signal Processing , 66 0 (8): 0 2041--2054, 2018. 10.1109/TSP.2018.2795539.DOI
  8. Petter Br ä nd é n. Solutions to two problems on permanents. Linear Algebra and its Applications , 436 0 (1): 0 53--58, 2012. 10.1016/j.laa.2011.06.022.DOI
  9. Zeineb Chebbi and Maher Moakher. Means of H ermitian positive-definite matrices based on the log-determinant α\alpha-divergence function. Linear Algebra and its Applications , 436 0 (7): 0 1872--1889, 2012. 10.1016/j.laa.2011.12.003.DOI
  10. Anoop Cherian, Suvrit Sra, Arindam Banerjee, and Nikolaos Papanikolopoulos. Jensen-- B regman logdet divergence with application to efficient similarity search for covariance matrices. IEEE Transactions on Pattern Analysis and Machine Intelligence , 35 0 (9): 0 2161--2174, 2013. 10.1109/TPAMI.2012.259.DOI
Show all 46 referencesShow fewer references
  1. Harry Crane. Some algebraic identities for the α\alpha-permanent. Linear Algebra and its Applications , 439 0 (11): 0 3445--3459, 2013. 10.1016/j.laa.2013.09.028.DOI
  2. Michel Marie Deza and Elena Deza. Encyclopedia of Distances . Springer, Berlin, 2nd edition, 2013. 10.1007/978-3-642-30958-8.DOI
  3. Jacques Faraut and Adam Kor á nyi. Function spaces and reproducing kernels on bounded symmetric domains. Journal of Functional Analysis , 88 0 (1): 0 64--89, 1990. 10.1016/0022-1236(90)90119-6.DOI
  4. Jacques Faraut and Adam Kor á nyi. Analysis on Symmetric Cones . Oxford Mathematical Monographs. Clarendon Press, Oxford, 1994. ISBN 978-0-19-853477-8.
  5. Carl H. FitzGerald and Roger A. Horn. On fractional H adamard powers of positive definite matrices. Journal of Mathematical Analysis and Applications , 61 0 (3): 0 633--642, 1977. 10.1016/0022-247X(77)90167-6.DOI
  6. Dominique Foata and Doron Zeilberger. Laguerre polynomials, weighted derangements, and positivity. SIAM Journal on Discrete Mathematics , 1 0 (4): 0 425--433, 1988. 10.1137/0401043.DOI
  7. P é ter E. Frenkel. Remarks on the α\alpha-permanent. Mathematical Research Letters , 17 0 (4): 0 795--802, 2010. 10.4310/MRL.2010.v17.n4.a17.DOI
  8. William Fulton and Joe Harris. Representation Theory: A First Course . Graduate Texts in Mathematics 129. Springer, New York, 1991.
  9. Simon G. Gindikin. Analysis in homogeneous domains. Russian Mathematical Surveys , 19 0 (4): 0 1--89, 1964. 10.1070/RM1964v019n04ABEH001153.DOI
  10. I. P. Goulden and D. M. Jackson. Immanants, S chur functions, and the M ac M ahon master theorem. Proceedings of the American Mathematical Society , 115 0 (3): 0 605--612, 1992. 10.2307/2159206.DOI
  11. Loo-Keng Hua. Inequalities involving determinants. Acta Mathematica Sinica , 5: 0 463--470, 1955.
  12. Shuangzhe Liu and G ö tz Trenkler. Hadamard, K hatri-- R ao, K ronecker and other matrix products. International Journal of Information and Systems Sciences , 4 0 (1): 0 160--177, 2008.
  13. Marvin Marcus. On a determinantal inequality. The American Mathematical Monthly , 65: 0 266--268, 1958. 10.2307/2310245.DOI
  14. Marvin Marcus and William Watkins. Partitioned H ermitian matrices. Duke Mathematical Journal , 38 0 (2): 0 237--249, 1971. 10.1215/S0012-7094-71-03830-0.DOI
  15. Russell Merris. Multilinear Algebra . Gordon and Breach, Amsterdam, 1997.
  16. Frank Nielsen. The S iegel-- K lein disk: H ilbert geometry of the S iegel disk domain. Entropy , 22 0 (9): 0 1019, 2020. 10.3390/e22091019.DOI
  17. A. Yu. Orlov. New solvable matrix integrals. International Journal of Modern Physics A , 19 0 (supp02): 0 276--293, 2004. 10.1142/S0217751X04020476.DOI
  18. A. M. Ostrowski and Olga Taussky. On the variation of the determinant of a positive definite matrix. Indagationes Mathematicae (Proceedings) , 54: 0 383--385, 1951. 10.1016/s1385-7258(51)50055-0.DOI
  19. S. Yu. Rotfel'd. The singular values of the sum of completely continuous operators. In M. Sh. Birman, editor, Topics in Mathematical Physics, Volume 3: Spectral Theory , pages 73--78. Consultants Bureau, New York, 1969.
  20. Alexander D Scott and Alan D Sokal. Complete monotonicity for inverse powers of some combinatorially defined polynomials. Acta Mathematica , 213: 0 323--392, 2014.
  21. Tomoyuki Shirai. Remarks on the positivity of α\alpha-determinants. Kyushu Journal of Mathematics , 61 0 (1): 0 169--189, 2007. 10.2206/kyushujm.61.169.DOI
  22. Suvrit Sra. On the matrix square root via geometric optimization. Electronic Journal of Linear Algebra , 31: 0 433--443, 2016 a . 10.13001/1081-3810.3196.DOI
  23. Suvrit Sra. Positive definite matrices and the S -divergence. Proceedings of the American Mathematical Society , 144 0 (7): 0 2787--2797, 2016 b . 10.1090/proc/12953.DOI
  24. Suvrit Sra. Metrics induced by J ensen-- S hannon and related divergences on positive definite matrices. Linear Algebra and its Applications , 616: 0 125--138, 2021. 10.1016/j.laa.2020.12.023.DOI
  25. Suvrit Sra and Reshad Hosseini. Conic geometric optimization on the manifold of positive definite matrices. SIAM Journal on Optimization , 25 0 (1): 0 713--739, 2015. 10.1137/140978168.DOI
  26. Gabor Szeg ő . Ü ber gewisse P otenzreihen mit lauter positiven K oeffizienten. Mathematische Zeitschrift , 37 0 (1): 0 674--688, 1933. 10.1007/bf01474608.DOI
  27. Gang Tian. Canonical Metrics in K ähler Geometry . Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 2000.
  28. Malik Tiomoko, Romain Couillet, Florent Bouchard, and Guillaume Ginolhac. Random matrix improved covariance estimation for a large class of metrics. In Proceedings of the 36th International Conference on Machine Learning , Proceedings of Machine Learning Research 97, pages 6254--6263, 2019.
  29. Mitsuru Uchiyama. Subadditivity of eigenvalue sums. Proceedings of the American Mathematical Society , 134 0 (5): 0 1405--1412, 2006. 10.1090/s0002-9939-05-08116-5.DOI
  30. David Vere-Jones. A generalization of permanents and determinants. Linear Algebra and its Applications , 111: 0 119--124, 1988. 10.1016/0024-3795(88)90053-5.DOI
  31. D á niel Virosztek. The metric property of the quantum J ensen-- S hannon divergence. Advances in Mathematics , 380: 0 107595, 2021. 10.1016/j.aim.2021.107595.DOI
  32. Changqing Xu, Zhaodi Xu, and Fuzhen Zhang. Revisiting H ua-- M arcus-- B ellman-- A ndo inequalities on contractive matrices. Linear Algebra and its Applications , 430 0 (5--6): 0 1499--1508, 2009. 10.1016/j.laa.2007.11.011.DOI
  33. Guanghui Xu, Changqing Xu, and Fuzhen Zhang. Contractive matrices of H ua type. Linear and Multilinear Algebra , 59 0 (2): 0 159--172, 2011. 10.1080/03081080903266888.DOI
  34. Florian Yger, Maxime Berar, and Fabien Lotte. Riemannian approaches in brain-computer interfaces: a review. IEEE Transactions on Neural Systems and Rehabilitation Engineering , 25 0 (10): 0 1753--1762, 2017. 10.1109/TNSRE.2016.2627016.DOI
  35. Artjom Zern, Matthias Zisler, Freddie Åstr ö m, Stefania Petra, and Christoph Schn ö rr. Unsupervised label learning on manifolds by spatially regularized geometric assignment. In Pattern Recognition (GCPR 2018) , Lecture Notes in Computer Science 11269, pages 698--713. Springer, 2019. 10.1007/978-3-030-12939-2_48.DOI
  36. Fuzhen Zhang. Recent results and research problems on H ua matrices. NIU Linear Algebra Meeting, presentation slides, August 2009.

Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 28, 2026