×

zbMATH — the first resource for mathematics

Majorizations and inequalities in matrix theory. (English) Zbl 0798.15024
This article is an excellent survey of results related to majorization in matrix theory since the appearance of the book “Inequalities: theory of majorization and its applications” (1979; Zbl 0437.26007) by A. W. Marshall and I. Olkin. This survey consists of eight sections: 1) Majorization for sequences, 2) Majorization for matrices, 3) Matrix means, 4) Matrix inequalities, 5) Log majorization, 6) Spectral perturbation, 7) Hadamard products and 8) Majorizations in von Neumann algebras. A great number of recent references are also given.

MSC:
15A45 Miscellaneous inequalities involving matrices
15A42 Inequalities involving eigenvalues and eigenvectors
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Akemann, C.A.; Anderson, J.; Pedersen, G.K., Triangular inequalities in operator algebras, Linear and multilinear algebra, Mr, 84a, 46126-178, (1982)
[2] Alberti, P.M.; Uhlmann, A., Stochasticity and partial order, (), 46057a-46057b
[3] Alfsen, E.M., Compact convex sets and boundary integrals, (1971), Springer-Verlag New York, MR 56#3615. · Zbl 0209.42601
[4] Amir-Moéz, A.R., Extreme properties of linear transformations and geometry in unitary spaces, (1968), Texas Tech. Univ Lubbock, Tex, MR41#1751. · Zbl 0188.07603
[5] Anderson, W.N.; Duffin, R.J., Series and parallel addition of matrices, J. math. anal. appl., 26, 576-594, (1969), MR 39#3904. · Zbl 0177.04904
[6] Anderson, W.N.; Morley, T.D.; Trapp, G.E., Symmetric function means of positive operators, Linear algebra appl., Mr, 85j, 47015-143, (1984), MR · Zbl 0575.47001
[7] Anderson, W.N.; Morley, T.D.; Trapp, G.E., Set of positive operators with suprema, SIAM J. matrix anal. appl., Mr, 91b, 47037-211, (1990)
[8] Ando, T., Topics on operator inequalities, (1979), Hokkaido Univ Sapporo, Japan, MR 58#2451. · Zbl 0388.47024
[9] Ando, T., Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear algebra appl., Mr, 80f, 15023-241, (1979) · Zbl 0495.15018
[10] Ando, T., Hua-Marcus inequalities, Linear and multilinear algebra, Mr, 81b, 15013-352, (1979) · Zbl 0438.15019
[11] Ando, T., An inequality between symmetric function means of positive operators, Acta sci. math. (Szeged), Mr, 85b, 47014-22, (1983) · Zbl 0532.47015
[12] Ando, T., Totally positive matrices, Linear algebra appl., Mr, 88b, 15023-219, (1987) · Zbl 0613.15014
[13] Ando, T., On some operator inequalities, Math. ann., Mr, 89c, 47019-159, (1987) · Zbl 0613.47020
[14] Ando, T., Comparison of norms ⦀f(A) − f(B)⦀ and ⦀f(|A − B|)⦀, Math. Z., Mr, 90a, 47021-409, (1988) · Zbl 0618.47007
[15] Ando, T., Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear algebra appl., Mr, 90g, 15034-248, (1989) · Zbl 0673.15011
[16] Ando, T., Parametrization of minimal points of some convex sets of matrices, Acta sci. math. (Szeged), 57, 3-10, (1993) · Zbl 0792.15013
[17] Ando, T., Matrix Young inequalities, Oper. theory adv. appl., (1994), to appear. · Zbl 0830.47010
[18] Ando, T., A majorization relation for Hadamard product of positive definite matrices, (1994), to appear.
[19] Ando, T.; Bhatia, R., Eigenvalue inequalities associated with the Cartesian decomposition, Linear and multilinear algebra, Mr, 89c, 15022-147, (1989) · Zbl 0641.15007
[20] Ando, T.; Hiai, F., Log-majorization and complementary Golden-Thompson type inequalities, Linear algebra appl., (1993), to appear. · Zbl 0793.15011
[21] Ando, T.; Hiai, F., Inequalities between powers of positive semidefinite matrices, Linear algebra appl., (1994), to appear. · Zbl 0817.15011
[22] Ando, T.; Horn, R.; Johnson, C., The singular values of a Hadamard product: basic inequalities, Linear and multilinear algebra, Mr, 89g, 15018-364, (1987)
[23] Ando, T.; Kubo, F., Some matrix inequalities in multiport network connections, Oper. theory adv. appl., Mr, 91d, 15038-131, (1989)
[24] Ando, T.; Kubo, F., Inequalities among operator symmetric function means, Progr. systems control theory, Mr, 92m, 47035-542, (1990)
[25] Ando, T.; Nakamura, Y., Some extremal problems related to majorization, Oper. theory adv. appl., Mr, 92f, 46026-92, (1991) · Zbl 0736.46008
[26] Ando, T.; Okubo, K., Induced norms of the Schur multiplication operators, Linear algebra appl., Mr, 92k, 47015-199, (1991) · Zbl 0722.15024
[27] Angelos, J.R.; Cowen, C.C.; Narayan, S.K., Triangular truncation and finding the norm of a Hadamard multiplier, Linear algebra appl., (1993), to appear. · Zbl 0751.15010
[28] Araki, H., On an inequality of Lieb and Thirring, Lett. math. phys., Mr, 91d, 47020-1170, (1990)
[29] Araki, H.; Yamagami, S., An inequality for the Hilbert-Schmidt norm, Comm. math. phys., Mr, 82j, 46076-91, (1981)
[30] Bapat, R.B., Majorization and singular values, Linear and multilinear algebra, Mr, 89j, 15015-214, (1987) · Zbl 0651.15014
[31] Bapat, R.B., Majorization and singular values II, SIAM J. matrix anal. appl., Mr, 90i, 15009-434, (1989) · Zbl 0686.15006
[32] Bapat, R.B., Majorization and singular values III, Linear algebra appl., Mr, 91k, 15018-70, (1991) · Zbl 0741.15009
[33] Bapat, R.B.; Kwong, A generalization of A ∘ A-1 ⩾ I, Linear algebra appl., Mr, 88g, 15014-111, (1987)
[34] Bapat, R.B.; Sunder, V.S., On majorization and Schur products, Linear algebra appl., Mr, 87c, 15018-117, (1985) · Zbl 0577.15016
[35] Bernstein, D.S., Inequalities for the trace of matrix exponentials, SIAM J. matrix anal. appl., Mr, 89b, 15031-158, (1988) · Zbl 0658.15018
[36] Bhagwat, K.V.; Subramanian, R., Inequalities between means of positive operators, Math. proc. Cambridge philos. soc., 83, 393-401, (1978), MR 67#7231. · Zbl 0375.47017
[37] Bhatia, R., Analysis of spectral variation and some inequalities, Trans. amer. math. soc., Mr, 83k, 15015-331, (1982)
[38] Bhatia, R., The distance between the eigenvalues of Hermitian matrices, Proc. amer. math. soc., Mr, 87b, 15021-42, (1986)
[39] Bhatia, R., Perturbation bounds for matrix eigenvalues, (), 15020
[40] Bhatia, R., Perturbation inequalities for the absolute value map in norm ideals of operator, J. operator theory, Mr, 89g, 47010-136, (1988)
[41] Bhatia, R.; Bhattacharyya, T., A generalization of the hoffman-Wielandt theorem, Linear algebra appl., 179, 11-17, (1993) · Zbl 0774.15011
[42] Bhatia, R.; Davis, C., A bound for the spectral variation of a unitary operator, Linear and multilinear algebra, Mr, 85b, 15020-76, (1984)
[43] Bhatia, R.; Davis, C., Concavity of certain functions of matrices, Linear and multilinear algebra, Mr, MR 87j, 15051-164, (1985)
[44] Bhatia, R.; Davis, C., More matrix forms of the arithmetic geometric Mean inequality, SIAM J. matrix anal. appl., 14, 132-136, (1993) · Zbl 0767.15012
[45] Bhatia, R.; Davis, C.; Koosis, P., An extremal problem in Fourier analysis with application to operator theory, J. funct. anal., Mr, 91a, 42006-150, (1989)
[46] Bhatia, R.; Davis, C.; McIntosh, A., Perturbation of spectral subspaces and solutions of linear operator equations, Linear algebra appl., Mr, 85a, 47020-67, (1983)
[47] Bhatia, R.; Holbrook, J.A., Short normal paths and spectral variation, Proc. amer. math. soc., Mr, 86e, 15017-282, (1985)
[48] Bhatia, R.; Holbrook, J.A., A softer, stronger lidskii theorem, Proc. Indian acad. sci., Mr, 90h, 15034-83, (1989) · Zbl 0675.15001
[49] Bhatia, R.; Kittaneh, F., On the singular values of a product of operators, SIAM J. matrix anal. appl., Mr, 90m, 47033-277, (1990)
[50] Birman, M.Sh.; Koplyanko, L.S.; Solomyak, M.Z., Estimate of the spectrum of the difference between fractional powers of self-adjoint operators, Izv. vyssh. uchebn. zaved. mat., 19, 3-10, (1975), MR 52#6458. · Zbl 0313.47020
[51] Chan, N.N.; Kwong, K., Hermitian matrix inequalities and a conjecture, Amer. math. monthly, Mr, 87d, 15011-541, (1985) · Zbl 0587.15009
[52] Cohen, J.E., Spectral inequalities for matrix exponentials, Linear algebra appl., Mr, 90e, 15025-28, (1988) · Zbl 0662.15012
[53] Corach, G.; Porta, H.; Recht, L., An operator inequality, Linear algebra appl., Mr, 91m, 47020-158, (1990) · Zbl 0733.47009
[54] Cowen, C.C.; Dritschel, M.A.; Penney, R.C., Norms of Hadamard multipliers, SIAM J. matrix anal. appl., (1993), to appear. · Zbl 0793.15017
[55] Davidson, K.R., The distance between unitary orbits of normal operators, Acta sci. math. (Szeged), Mr, 88c, 47040-223, (1986) · Zbl 0614.47018
[56] Davies, E.B., Lipschitz continuity of functions of operators in the Schatten classes, J. London math. soc., Mr, 89c, 47009-157, (1988) · Zbl 0648.47011
[57] Davis, C., Notions generalizing convexity for functions defined on spaces of matrices, (), 187-201, MR 27#5771
[58] Dodds, P.G.; Dodds, T.K., On a submajorization inequality of T. Ando, Oper. theory adv. appl., (1994), to appear. · Zbl 0826.46054
[59] Dodds, P.G.; Dodds, T.K.; de Pagter, B., Non-commutative Banach function spaces, Math. Z., M.r., 90j, 46054-597, (1989) · Zbl 0653.46061
[60] Donoghue, W., Monotone matrix functions and analytic continuation, (1974), Springer-Verlag New York, MR 58#6279 · Zbl 0278.30004
[61] Elsner, L., A note on the hoffman-Wielandt theorem, Linear algebra appl., 182, 235-237, (1993) · Zbl 0803.15023
[62] Fack, T., Sur la notion de valeur caractéristique, J. operator theory, Mr, 84m, 47012-333, (1982) · Zbl 0493.46052
[63] Fack, T.; Kosaki, H., Generalized s-numbers of τ-measurable operators, Pacific J. math., Mr, 87h, 46122-300, (1986) · Zbl 0617.46063
[64] Fiedler, M., Über eine ungleichung fur positive definite matrizen, Math. nach., 23, 197-199, (1961), MR 25#3049. · Zbl 0111.01503
[65] Fiedler, M., A note on the Hadamard product of matrices, Linear algebra appl., Mr, 84c, 15025-235, (1983)
[66] Fisher, P.; Holbrook, A.J.R., Matrices sous-stochastiques et fonctions convexes, Canad. J. math., 29, 631-637, (1977), MR 58#30042. · Zbl 0344.46002
[67] Fisher, P.; Holbrook, A.J.R., Balayage defined by the non-negative convex functions, Proc. amer. math. soc., Mr, 81f, 46012-448, (1980)
[68] Friedland, S.; Katz, M., On a matrix inequality, Linear algebra appl., Mr, 87m, 15053-190, (1987)
[69] Fujii, M.; Furuta, T.; Kamei, E., Furuta’s inequality and its application to Ando’s theorem, Linear algebra appl., 179, 161-169, (1993) · Zbl 0788.47012
[70] Furuta, T., A ⩾ B ⩾ 0 assures (brapbr)\(1p\) ⩾ B\((p+2r)q\) for r ⩾ 0, p ⩾ 0, q ⩾ 1 with (1 + 2r)q ⩾ (p + 2r), Proc. amer. math. soc., Mr, 89b, 47023-88, (1987) · Zbl 0721.47023
[71] Furuta, T., Norm inequalities equivalent to Loewner-Heinz theorem, Rev. math. phys., Mr, 91b, 47028-137, (1989)
[72] Furuta, T., Extension of the Furuta inequality and log majorization by Ando and Hiai, Linear algebra appl., (1994), to appear. · Zbl 0822.15008
[73] Garloff, J., Majorization between the diagonal elements and the eigenvalues of an oscillating matrix, Linear algebra appl., Mr, 83m, 15012-184, (1982) · Zbl 0493.15016
[74] Garloff, J., An inverse eigenvalue problem for totally nonnegative matrices, Linear and multilinear algebra, Mr, 87i, 15009-23, (1985) · Zbl 0557.15010
[75] Gil, M.I., On inequalities for eigenvalues of matrices, Linear algebra appl., 184, 201-206, (1993) · Zbl 0773.15007
[76] Gohberg, I.; Krein, M.G., Introduction to the theory of linear nonselfadjoint operators, (1965), Amer. Math. Soc Providence, MR 36#3137. · Zbl 0181.13504
[77] Gohberg, I.; Krein, M.G., Theory of Volterra operators in Hilbert space and its applications (transl.), (1967), Amer. Math. Soc Providence, MR #362007.
[78] Hansen, F., An operator inequality, Math. ann., Mr, 82a, 46065-250, (1980)
[79] Hansen, F.; Pedersen, G.K., Jensen’s inequality for operators and Loewner’s theorem, Math. ann., Mr, 83g, 47020-241, (1981/82)
[80] Hässelbarth, W.; Ruch, E., On the extension of linear contractions, Linear algebra appl., (1993), to appear. · Zbl 0771.15009
[81] Hersch, J.; Zwahlen, B.P., Évaluations par default pour une somme quelconque de valeurs propres γ_k d’un opérateur C = A + B a l’aide de valeurs propres αi de A et βj de B, C. R. acad. sci. Paris, 254, 1559-1561, (1962), MR 24#A2583. · Zbl 0112.01502
[82] Hiai, F., Majorization and stochastic maps in von Neumann algebras, J. math. anal. appl., Mr, 88k, 46076-48, (1987) · Zbl 0634.46051
[83] Hiai, F.; Nakamura, Y., Majorizations for generalized s-numbers in semifinite von Neumann algebras, Math. Z., Mr, 88g, 46070-27, (1987) · Zbl 0598.46039
[84] Hiai, F.; Nakamura, Y., Distance between unitary orbits in von Neumann algebra, Pacific J. math., Mr, 90i, 46105-294, (1989)
[85] Hiai, F.; Petz, D., The Golden-Thompson trace inequality is complemented, Linear algebra appl., 181, 153-185, (1993) · Zbl 0784.15011
[86] Holbrook, A.J., Spectral variation of normal matrices, Linear algebra appl., 174, 131-144, (1992) · Zbl 0761.15017
[87] Horn, A., Eigenvalues of sum of Hermitian matrices, Pacific J. math., 9, 225-241, (1962), MR 15#847. · Zbl 0112.01501
[88] Horn, R.; Johnson, C.; Johnson, C., The Hadamard product, Matrix theory and applications, Proc. symp. appl. math., Mr, 92a, 15001-169, (1990)
[89] Horn, R.; Johnson, C., Hadamard and conventional submultiplicativity for unitarily invariant norms on matrices, Linear and multilinear algebra, Mr, 89b, 15037-106, (1987) · Zbl 0605.15017
[90] Hua, L., Inequalities involving determinants, Acta math. sinica, 5, 463-470, (1955), MR 17#703. · Zbl 0066.26601
[91] Johnson, C.; Elsner, L., The relation between the Hadamard and conventional multiplication for positive definite matrices, Linear algebra appl., Mr, 88i, 15039-240, (1987)
[92] Kamei, E., Majorization in finite factors, Math. japon., Mr, 84g, 46086-499, (1983) · Zbl 0527.47016
[93] Kamei, E., Doubly stochasticity in finite factors, Math. japon., Mr, 88a, 46067a-907, (1984)
[94] Kamei, E., An order on statistical operators implicitly introduced by von Neumann, Math. japon., Mr, 87k, 47048-443, (1985)
[95] Kato, T., Continuity of the map S → |S| for linear operators, Proc. Japan acad., 49, 157-160, (1973), MR 53#8943. · Zbl 0301.47006
[96] Kato, T., Spectral order and a matrix limit theorem, Linear and multilinear algebra, Mr, 88j, 47030-19, (1979)
[97] Kittaneh, F., Inequalities for the Schatten p-norm III, Comm. math. phys., Mr, 88j, 47009-310, (1986) · Zbl 0595.47012
[98] Kittaneh, F., Inequalities for the Schatten p-norm IV, Comm. math. phys., Mr, 88j, 47010-310, (1986)
[99] Kittaneh, F., A note on the arithmetic—geometric-Mean inequality for matrices, Linear algebra appl., 171, 1-8, (1992) · Zbl 0755.15008
[100] Kittaneh, F., Norm inequalities for fractional powers of positive operators, Lett. math. phys., 27, 279-285, (1993) · Zbl 0895.47003
[101] Kittaneh, F., On some operator inequalities, (1994), to appear. · Zbl 0803.47019
[102] Kittaneh, F.; Kosaki, H., Inequalities for the Schatten p-norm V, Publ. res. inst. math. sci., Mr, 88h, 47011-443, (1987) · Zbl 0627.47002
[103] Komiya, H., Necessary and sufficient conditions for multivariate majorization, Linear algebra appl., Mr, 84k, 15015-154, (1983) · Zbl 0526.15017
[104] Kosaki, H. 1983. Private communication.
[105] Kosaki, H., On the continuity of the map φ → |φ| from the predual of a W∗-algebra, J. funct. anal., Mr, 86c, 46072-131, (1984) · Zbl 0584.46050
[106] Kosaki, H., An inequality of Araki-Lieb-Thirring, Proc. amer. math. soc., Mr, 92g, 46079-481, (1992) · Zbl 0762.46060
[107] Kosaki, H., Unitarily invariant norm under which the map A ↦ |A| is Lipschitz continuous, Publ. res. inst. math. sci., Mr, 93d, 47043-313, (1992)
[108] Kubo, F.; Ando, T., Means of positive linear operators, Math. ann., Mr, 84d, 47028-224, (1979/80) · Zbl 0412.47013
[109] Li, C.K., A generalization of spectral radius, numerical radius, and spectral norm, Linear algebra appl., Mr, 88g, 26015-118, (1987) · Zbl 0624.15012
[110] Li, C.K.; Tsing, N.K., Distance to the convex hull of unitary orbits, Linear and multilinear algebra, Mr, 90k, 15011-103, (1989)
[111] Lidskii, B.V., A polyhedron of the spectrum of the sum of two Hermitian matrices, Funktsional. anal. i prilozhen, Mr, 83k, 15009-77, (1982)
[112] Lieb, E., Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv. in math., 11, 267-288, (1973), MR 48#6218. · Zbl 0267.46055
[113] Marcus, M.; Kidman, K.; Sandy, K., Unitarily invariant generalized matrix norms and Hadamard products, Linear and multilinear algebra, Mr, 87b, 15039-213, (1984) · Zbl 0563.15017
[114] Marcus, M.; Lopes, L., Inequalities for symmetric functions and Hermitian matrices, Canad. J. math., 9, 305-312, (1957), MR 18#877. · Zbl 0079.02103
[115] Marcus, A.S., Eigenvalues and singular values of the sum and product of linear operators, Russian math. surveys, 19, 91-120, (1964), MR 24#6318. · Zbl 0133.07205
[116] Marshall, A.W.; Olkin, I., Inequalities: theory of majorization and its applications, (), 00002
[117] Marshall, A.W.; Olkin, I., Inequalities for the trace function, Aequationes math., Mr, 87e, 15037-39, (1985)
[118] Mathias, R., Concavity of monotone matrix function of finite order, Linear and multilinear algebra, Mr, 91f, 47011-138, (1990)
[119] Merris, R., Extension of the Minkowski determinant theorem, Portugal. math., Mr, 84c, 15009-153, (1982)
[120] Nakamura, Y., An inequality for generalized s-numbers, Integral equations operator theory, Mr, 88f, 47016-145, (1987)
[121] Nakamura, Y., Classes of operator monotone functions and Stieltjes functions, Oper. theory adv. appl., Mr, 91a, 47017-404, (1989)
[122] Nudelman, A.A.; Shvartzman, P.A., On the spectra of a product of unitary matrices, Uspekhi mat. nauk, 13, 84, 111-117, (1959), MR 21#3436.
[123] Okubo, K., Hölder-type norm inequalities for Schur products of matrices, Linear algebra appl., Mr, 88f, 15028-28, (1987) · Zbl 0633.15013
[124] Olson, N.P., The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice, Proc. amer. math. soc., 28, 537-544, (1971), MR 43#2528. · Zbl 0215.20504
[125] Ovchinikov, V.I., s-number of measurable operators, Funktsional. anal. i prilozhen, 4, 236-242, (1970), MR 42#6644.
[126] Paulsen, V.I.; Power, S.C.; Smith, R.R., Schur products and matrix completions, J. funct. anal., Mr, 90j, 46051-178, (1989) · Zbl 0672.15008
[127] Petz, D., Spectral scale of selfadjoint operators and trace inequalities, J. math. anal. appl., Mr, c, 47055-82, (1985) · Zbl 0655.47032
[128] Pusz, W.; Woronowicz, S.L., Functional calculus for sesquilinear forms and the purification map, Rep. math. phys., 8, 159-170, (1975), MR 54#8316. · Zbl 0327.46032
[129] Riddle, R.C., Minimax problems on Grassmann manifolds, sums of eigenvalues, Adv. math., 54, 107-199, (1984), MR 86#58019 · Zbl 0599.58016
[130] Ruch, E.; Schranner, R.; Seligman, R., Generalization of a theorem of Hardy-Littlewood-polya, J. math. anal. appl., Mr, 81m, 26015-229, (1980)
[131] Seiler, E.; Simon, B., An inequality among determinants, Proc. nat. acad. sci. U.S.A., 72, 3277-3278, (1975), MR 55#6225. · Zbl 0313.47011
[132] Simon, B., Trace ideals and their applications, (), 47048
[133] Smiley, M.F., Inequalities related to Lidskii’s, Proc. amer. math. soc., 19, 1029-1034, (1966), MR 38#2154. · Zbl 0169.35103
[134] Sonis, M.G., On a class of operators in von Neumann algebras with Segal measure on the projections, Math. USSR sb., 13, 344-357, (1971), MR 46#681. · Zbl 0238.46061
[135] Sunder, V.S., On permutations, convex hulls and normal operators, Linear algebra appl., Mr, 83c, 47040-411, (1982) · Zbl 0537.15014
[136] Sunder, V.S., Distance between normal operators, Proc. amer. math. soc., Mr, 83c, 47040-484, (1982) · Zbl 0486.47015
[137] Takesaki, M., Theory of operator algebras I, (), 46038
[138] Thompson, C.J., Inequalities and partial orders on matrix spaces, Indiana univ. math. J., 21, 469-480, (1971), Mr 45#3442. · Zbl 0227.15005
[139] Thompson, R.C., A matrix inequality, Comment. math. univ. carolin., 17, 376-393, (1976), MR 54#7503. · Zbl 0339.15008
[140] Thompson, R.C., Matrix type metric inequalities, Linear and multilinear algebra, 5, 303-319, (1978), MR 58#16725. · Zbl 0433.15010
[141] Thompson, R.C., The case of equality in the matrix valued triangle inequality, Pacific J. math., 82, 279-280, (1979) · Zbl 0412.15013
[142] Thompson, R.C., Proof of a conjectured exponential formula, Linear and multilinear algebra, Mr, 88b, 15027-189, (1986)
[143] Trapp, G.E., Hermitian semidefinite matrix means and related inequalities—an introduction, Linear and multilinear algebra, Mr, 86b, 15009-123, (1984) · Zbl 0548.15013
[144] Walter, M.E., On the norm of a Schur product, Linear algebra appl., Mr, 87j, 15048-213, (1986)
[145] Wang, B.Y.; Gong, M.P., Some eigenvalue inequalities for positive semidefinite matrix power products, Linear algebra appl., 184, 249-260, (1993) · Zbl 0773.15009
[146] Zhang, F., Another proof of a singular value inequality concerning Hadamard products of matrices, Linear and multilinear algebra, Mr, 89f, 15022-311, (1987)
[147] Zhang, F., A majorization conjecture for Hadamard products and compound matrices, Linear and multilinear algebra, 33, 301-303, (1993)
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.