Majorization, doubly stochastic matrices, and comparison of eigenvalues
Open accessThe claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
A function M of Schmidt coefficients is Schur concave if and only if it is invariant under permutations of any two arguments and satisfies Eq. (24).
“It is satisfied if and only if [48] M given as a function of the Schmidt coefficients is invariant under permutations of any two arguments and satisfies”
From Measures and dynamics of entangled states (2005) · cites this paper as [48]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
Inequalities: Theory of Majorization and its Applications.
Arnold, B. C., Marshall, A. W., & Olkin, I. (1981) · Journal of the American Statistical Association · doi:10.2307/2287859 · Cited by 4,989 · see citing works
Ford, D. R., & Fulkerson, D. R. (1962) · Cited by 4,080 · see citing works
Theory of Operator Algebras II
Takesaki, M. (2003) · Encyclopaedia of mathematical sciences · doi:10.1007/978-3-662-10451-4 · Cited by 3,257 · see citing works
Inequalities - Theory of Majorization and Its Applications
(1979) · Mathematics in Science and Engineering/Mathematics in science and engineering · doi:10.1016/c2009-0-22048-4 · Cited by 2,890 · see citing works
Non-negative Matrices and Markov Chains
Seneta, E. B. (1981) · Springer series in statistics · doi:10.1007/0-387-32792-4 · Cited by 2,858 · see citing works
Completely positive linear maps on complex matrices
Choi, M.-D. (1975) · Linear Algebra and its Applications · doi:10.1016/0024-3795(75)90075-0 · Cited by 2,659 · see citing works
Introduction to the theory of linear nonselfadjoint operators
Крейн, M. Г., & Gohberg, I. (1969) · American Mathematical Society eBooks · Cited by 2,198 · see citing works
Takesaki, M. (1979) · Springer eBooks · doi:10.1007/978-1-4612-6188-9 · Cited by 1,629 · see citing works
Introduction to the Theory of Linear Non-selfadjoint Operators
Gohberg, I. (1969) · Medical Entomology and Zoology · Cited by 1,510 · see citing works
Equivalent Comparisons of Experiments
Blackwell, D. (1953) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177729032 · Cited by 1,408 · see citing works
Positive functions on 𝐶*-algebras
Stinespring, W. F. (1955) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-1955-0069403-4 · Cited by 1,085 · see citing works
Quantum Physics: A Functional Integral Point of View
Gllmm, J., Jaffe, A. M., & Wilczek, F. A. (1982) · Physics Today · doi:10.1063/1.2914804 · Cited by 1,062 · see citing works
On a Theorem of Weyl Concerning Eigenvalues of Linear Transformations I
Fan, K. (1949) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.35.11.652 · Cited by 1,045 · see citing works
A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices
Sinkhorn, R. (1964) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177703591 · Cited by 1,032 · see citing works
Perron, O. (1907) · Mathematische Annalen · doi:10.1007/bf01449896 · Cited by 907 · see citing works
Concerning nonnegative matrices and doubly stochastic matrices
Sinkhorn, R., & Knopp, P. (1967) · Pacific Journal of Mathematics · doi:10.2140/pjm.1967.21.343 · Cited by 882 · see citing works
SYMMETRIC GAUGE FUNCTIONS AND UNITARILY INVARIANT NORMS
Mirsky, L. (1960) · The Quarterly Journal of Mathematics · doi:10.1093/qmath/11.1.50 · Cited by 835 · see citing works
Compact Convex Sets and Boundary Integrals
Alfsen, E. M. (1971) · doi:10.1007/978-3-642-65009-3 · Cited by 821 · see citing works
�ber monotone Matrixfunktionen
L�wner, K. (1934) · Mathematische Zeitschrift · doi:10.1007/bf01170633 · Cited by 766 · see citing works
Positive Functions on C ∗ -Algebras
Stinespring, W. F. (1955) · Proceedings of the American Mathematical Society · doi:10.2307/2032342 · Cited by 753 · see citing works
QUANTUM PHYSICS: A Functional Integral Point of View
Streater, R. F. (1982) · Bulletin of the London Mathematical Society · doi:10.1112/blms/14.6.562 · Cited by 689 · see citing works
Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix
Horn, A. (1954) · American Journal of Mathematics · doi:10.2307/2372705 · Cited by 514 · see citing works
1. A Certain Zero-sum Two-person Game Equivalent to the Optimal Assignment Problem
von Neumann, J. (1953) · Princeton University Press eBooks · doi:10.1515/9781400881970-002 · Cited by 497 · see citing works
Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen.
Schur, J. (1911) · Journal für die reine und angewandte Mathematik (Crelles Journal) · doi:10.1515/crll.1911.140.1 · Cited by 490 · see citing works
Maximum Properties and Inequalities for the Eigenvalues of Completely Continuous Operators.
Fan, K. (1951) · Proceedings of the National Academy of Sciences of the United States of America · doi:10.1073/pnas.37.11.760 · Cited by 488 · see citing works
Unzerlegbare, nicht negative Matrizen
Wielandt, H. (1950) · Mathematische Zeitschrift · doi:10.1007/bf02230720 · Cited by 431 · see citing works
Some simple inequalities satisfied by convex functions
Hardy, G. H. (1929) · Medical Entomology and Zoology · Cited by 412 · see citing works
Inequalities between the Two Kinds of Eigenvalues of a Linear Transformation
Weyl, H. (1949) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.35.7.408 · Cited by 408 · see citing works
Godwin, H. J., & MITRINOVIC, D. S. (1971) · Journal of the Royal Statistical Society Series A (General) · doi:10.2307/2344253 · Cited by 392 · see citing works
A general rearrangement inequality for multiple integrals
Brascamp, H. J., Lieb, É. H., & Luttinger, J. M. (1974) · Journal of Functional Analysis · doi:10.1016/0022-1236(74)90013-5 · Cited by 365 · see citing works
Probability Inequalities in Multivariate Distributions.
Eaton, M. L., & Tong, Y. L. (1982) · Journal of the American Statistical Association · doi:10.2307/2287749 · Cited by 348 · see citing works
Probability: Inequalities in Multivariate Distributions.
Bibby, J., & Tong, Y. L. (1981) · Journal of the Royal Statistical Society Series A (General) · doi:10.2307/2981818 · Cited by 200 · see citing works
Equimeasurable rearrangements of functions
Chong, K., & Rice, N. M. (1971) · Cited by 187 · see citing works
An Inequality for Rearrangements
Lorentz, G. G. (1953) · American Mathematical Monthly · doi:10.2307/2307574 · Cited by 171 · see citing works
The diagonal equivalence of a nonnegative matrix to a stochastic matrix
Brualdi, R. A., Parter, S. V., & Schneider, H. (1966) · Journal of Mathematical Analysis and Applications · doi:10.1016/0022-247x(66)90184-3 · Cited by 169 · see citing works
Über Matrizen aus nicht negativen Elementen
Frobenius, G. (2013) · Publication Server (Berlin-Brandenburg Academy of Sciences and Humanities) · doi:10.3931/e-rara-18865 · Cited by 169 · see citing works
The solution of van der Waerden's problem for permanents
Егорычев, Г. П. (1981) · Advances in Mathematics · doi:10.1016/0001-8708(81)90044-x · Cited by 168 · see citing works
On a Theorem of Weyl Concerning Eigenvalues of Linear Transformations
Fan, K. (1950) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.36.1.31 · Cited by 166 · see citing works
Amir‐Moez, A. R. (1956) · Duke Mathematical Journal · doi:10.1215/s0012-7094-56-02343-2 · Cited by 153 · see citing works
Schur, I. (1909) · Mathematische Annalen · doi:10.1007/bf01450045 · Cited by 143 · see citing works
Rado, R. (1952) · Journal of the London Mathematical Society · doi:10.1112/jlms/s1-27.1.1 · Cited by 136 · see citing works
Results and problems in the theory of doubly-stochastic matrices
Mirsky, L. (1963) · Probability Theory and Related Fields · doi:10.1007/bf00533407 · Cited by 124 · see citing works
Convex and concave functions of singular values of matrix sums
Thompson, R. C. (1976) · Pacific Journal of Mathematics · doi:10.2140/pjm.1976.66.285 · Cited by 121 · see citing works
An extremum property of sums of eigenvalues
Wielandt, H. (1955) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-1955-0067842-9 · Cited by 119 · see citing works
The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice
Olson, M. P. (1971) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-1971-0276788-1 · Cited by 114 · see citing works
On a Theorem of Hardy, Littlewood, Polya, and Blackwell
Sherman, S. (1951) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.37.12.826 · Cited by 112 · see citing works
Kraus, F. J. (1936) · Mathematische Zeitschrift · doi:10.1007/bf01180403 · Cited by 111 · see citing works
On the Singular Values of a Product of Completely Continuous Operators
Horn, A. (1950) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.36.7.374 · Cited by 106 · see citing works
THE EIGEN- AND SINGULAR VALUES OF THE SUM AND PRODUCT OF LINEAR OPERATORS
Markus, A. S. (1964) · Russian Mathematical Surveys · doi:10.1070/rm1964v019n04abeh001154 · Cited by 93 · see citing works
Über quadratische Formen mit reellen Koeffizienten
Fischer, E. (1905) · Monatshefte für Mathematik · doi:10.1007/bf01693781 · Cited by 77 · see citing works
Day, P. (1972) · Canadian Journal of Mathematics · doi:10.4153/cjm-1972-093-x · Cited by 58 · see citing works
Stochasticity and partial order
(1983) · Discrete Applied Mathematics · doi:10.1016/0166-218x(83)90113-0 · Cited by 57 · see citing works
A new rearrangement inequality for multiple integrals
Luttinger, J. M., & Friedberg, R. M. (1976) · Archive for Rational Mechanics and Analysis · doi:10.1007/bf00251862 · Cited by 54 · see citing works
The proper values of the sum and product of symmetric matrices
Likskii, V. B., & Forsythe, G. E. (1953) · doi:10.6028/nbs.rpt.2248 · Cited by 48 · see citing works
Notes on Egoritsjev's proof of the van der Waerden conjecture
Lint, J. H. (1981) · Linear Algebra and its Applications · doi:10.1016/0024-3795(81)90285-8 · Cited by 44 · see citing works
Generalization of a theorem by Hardy, Littlewood, and Pólya
Ruch, E., Schranner, R., & Seligman, T. H. (1980) · Journal of Mathematical Analysis and Applications · doi:10.1016/0022-247x(80)90075-x · Cited by 43 · see citing works
Rearrangement inequalities involving convex functions
London, D. (1970) · Pacific Journal of Mathematics · doi:10.2140/pjm.1970.34.749 · Cited by 39 · see citing works
Funktionen auf Produktr�umen mit vorgegebenen Marginal-Funktionen
Kellerer, H. G. (1961) · Mathematische Annalen · doi:10.1007/bf01470505 · Cited by 36 · see citing works
Dissipative motion in state spaces
Alberti, P. M., & Uhlmann, A. (1981) · Cited by 30 · see citing works
Knuth, D. E. (1981) · American Mathematical Monthly · doi:10.1080/00029890.1981.11995356 · Cited by 29 · see citing works
Spectral order and a matrix limit theorem
Kato, T. (1979) · Linear and Multilinear Algebra · doi:10.1080/03081087908817295 · Cited by 28 · see citing works
Ruderman, H. D. (1952) · American Mathematical Monthly · doi:10.2307/2307185 · Cited by 26 · see citing works
Matrix type metric inequalities
Thompson, R. C. (1978) · Linear and Multilinear Algebra · doi:10.1080/03081087808817211 · Cited by 22 · see citing works
An Induction Theorem for Rearrangements
Chong, K. (1976) · Canadian Journal of Mathematics · doi:10.4153/cjm-1976-019-4 · Cited by 22 · see citing works
Über Matrizen aus positiven Elementen : II
Frobenius, G. (2025) · Publication Server (Berlin-Brandenburg Academy of Sciences and Humanities) · Cited by 22 · see citing works
Knuth, D. E. (1981) · American Mathematical Monthly · doi:10.2307/2321472 · Cited by 17 · see citing works
Balayage defined by the nonnegative convex functions
Fischer, P., & Holbrook, J. (1980) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-1980-0567989-9 · Cited by 16 · see citing works
Inequalities for certain Classes of Convex Functions
Mirsky, L. (1959) · Proceedings of the Edinburgh Mathematical Society · doi:10.1017/s0013091500021969 · Cited by 16 · see citing works
Minc, H. (1971) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1971-0283002-4 · Cited by 15 · see citing works
Majorization of vectors and inequalities for convex functions
Mirsky, L. (1961) · Monatshefte für Mathematik · doi:10.1007/bf01307025 · Cited by 13 · see citing works
Existence and Density Theorems for Stochastic Maps on Commutative C*‐algebras
Alberti, P. M., & Uhlmann, A. (1980) · Mathematische Nachrichten · doi:10.1002/mana.19800970125 · Cited by 9 · see citing works
Spectral order preserving matrices and Muirhead’s theorem
Chong, K. M. (1974) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1974-0379780-9 · Cited by 7 · see citing works
Matrices Sous-Stochastiques et Fonctions Convexes
Fischer, P., & Holbrook, J. (1977) · Canadian Journal of Mathematics · doi:10.4153/cjm-1977-064-5 · Cited by 6 · see citing works
Grothendieck, A. (1956) · French digital mathematics library (Numdam) · Cited by 6 · see citing works
An induction principle for spectral and rearrangement inequalitities
Chong, K. M. (1974) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1974-0344396-7 · Cited by 5 · see citing works
An Induction Principle for Spectral and Rearrangement Inequalities
Chong, K. (1974) · Transactions of the American Mathematical Society · doi:10.2307/1997033 · Cited by 4 · see citing works
A general induction theorem for rearrangements of n-tuples
Chong, K. (1976) · Journal of Mathematical Analysis and Applications · doi:10.1016/0022-247x(76)90121-9 · Cited by 2 · see citing works
Flagged first, then by citations
Research works printed but not linked: 2
- E. Beckenbach and R. Bellman, Inequalities, 2nd ed., Springer, New York, 1965; MR31#5937.
- H. Mint, Permanents, Addison-Wesley, New York, 1978; MR8Od#15009.
Cited by
Sorted by citations
Measures and dynamics of entangled states
Mintert, F., Carvalho, A. R. R., Kuś, M., & Buchleitner, A. (2005) · Physics Reports · doi:10.1016/j.physrep.2005.04.006 · Cited by 420 · see citing works
Better bound on the exponent of the radius of the multipartite separable ball
Gurvits, L., & Barnum, H. (2005) · Physical Review A · doi:10.1103/physreva.72.032322 · Cited by 44 · see citing works
- Authors
- Ando, T.
- Venue
- Linear Algebra and its Applications
- Publisher
- Elsevier BV
- Published
- 1989-06
- Volume
- 118
- Pages
- 163-248
- ISSN
- 0024-3795 (ISSN-L)
- OpenAlex
- W1997314000
- Cited by
- 305
- References
- 77
- Citations by year
- 2026: 7 · 2025: 12 · 2024: 8 · 2023: 9 · 2022: 13 · 2021: 15 · 2020: 22 · 2019: 18 · 2018: 13 · 2017: 12 · 2016: 15 · 2015: 14 · 2014: 18 · 2013: 9 · 2012: 9
- FWCI
- 3.03
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Linear Algebra and its Applications (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Inequalities and Applications
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Advanced Combinatorial Mathematics
- Keywords
- majorization, doubly stochastic matrices, eigenvalue comparison
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review