Spectral order preserving matrices and Muirhead’s theorem

Chong, Kong Ming

doi:10.1090/S0002-9947-1974-0379780-9

Spectral order preserving matrices and Muirhead’s theorem
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Trans. Amer. Math. Soc. 200 (1974), 437-444 Request permission

Abstract:

In this paper, a characterization is given for matrices which preserve the Hardy-Littlewood-Pólya spectral order relation $\prec$ for $n$-vectors in ${R^n}$. With this characterization, a new proof is given for the classical Muirhead theorem and some Muirhead-type inequalities are obtained. Moreover, sufficient conditions are also given for matrices which preserve the Hardy-Littlewood-Pólya weak spectral order relation $\prec \prec$.

References

Equimeasurable rearrangements of functions with applications to analysis

Kong Ming Chong, Some extensions of a theorem of Hardy, Littlewood and Pólya and their applications, Canadian J. Math. 26 (1974), 1321–1340. MR 352377, DOI 10.4153/CJM-1974-126-1
Kong Ming Chong, Spectral orders, uniform integrability and Lebesgue’s dominated convergence theorem, Trans. Amer. Math. Soc. 191 (1974), 395–404. MR 369646, DOI 10.1090/S0002-9947-1974-0369646-2
Kong Ming Chong, Variation reducing properties of decreasing rearrangements, Canadian J. Math. 27 (1975), 330–336. MR 374353, DOI 10.4153/CJM-1975-040-2
Kong Ming Chong, An induction principle for spectral and rearrangement inequalitities, Trans. Amer. Math. Soc. 196 (1974), 371–383. MR 344396, DOI 10.1090/S0002-9947-1974-0344396-7

Doubly stochastic operators and rearrangement theorems

K. M. Chong and N. M. Rice, Equimeasurable rearrangements of functions, Queen’s Papers in Pure and Applied Mathematics, No. 28, Queen’s University, Kingston, Ont., 1971. MR 0372140

Inequalities

L. Mirsky, Results and problems in the theory of doubly-stochastic matrices, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1 (1962/63), 319–334. MR 153038, DOI 10.1007/BF00533407