Trace inequality – Wikipedia

In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices.^[1][2][3][4]

Basic definitions[edit]

Let

${displaystyle mathbf {H} _{n}}$

denote the space of Hermitian

${displaystyle ntimes n}$

matrices,

${displaystyle mathbf {H} _{n}^{+}}$

denote the set consisting of positive semi-definite

${displaystyle ntimes n}$

Hermitian matrices and

${displaystyle mathbf {H} _{n}^{++}}$

denote the set of positive definite Hermitian matrices. For operators on an infinite dimensional Hilbert space we require that they be trace class and self-adjoint, in which case similar definitions apply, but we discuss only matrices, for simplicity.

For any real-valued function

${displaystyle f}$

on an interval

${displaystyle Isubseteq mathbb {R} ,}$

one may define a matrix function

${displaystyle f(A)}$

for any operator

${displaystyle Ain mathbf {H} _{n}}$

with eigenvalues

${displaystyle lambda }$

in

${displaystyle I}$

by defining it on the eigenvalues and corresponding projectors

${displaystyle P}$

as

given the spectral decomposition

${displaystyle A=sum _{j}lambda _{j}P_{j}.}$

Operator monotone[edit]

A function

${displaystyle f:Ito mathbb {R} }$

defined on an interval

${displaystyle Isubseteq mathbb {R} }$

is said to be operator monotone if for all

${displaystyle n,}$

and all

${displaystyle A,Bin mathbf {H} _{n}}$

with eigenvalues in

${displaystyle I,}$

the following holds,

where the inequality

${displaystyle Ageq B}$

means that the operator

${displaystyle A-Bgeq 0}$

is positive semi-definite. One may check that

${displaystyle f(A)=A^{2}}$

is, in fact, not operator monotone!

Operator convex[edit]

A function

${displaystyle f:Ito mathbb {R} }$

is said to be operator convex if for all

${displaystyle n}$

and all

${displaystyle A,Bin mathbf {H} _{n}}$

with eigenvalues in

${displaystyle I,}$

and

${displaystyle 0$

, the following holds

Note that the operator

${displaystyle lambda A+(1-lambda )B}$

has eigenvalues in

${displaystyle I,}$

since

${displaystyle A}$

and

${displaystyle B}$

have eigenvalues in

${displaystyle I.}$

A function

${displaystyle f}$

is operator concave if

${displaystyle -f}$

is operator convex;=, that is, the inequality above for

${displaystyle f}$

is reversed.

Joint convexity[edit]

A function

${displaystyle g:Itimes Jto mathbb {R} ,}$

defined on intervals

${displaystyle I,Jsubseteq mathbb {R} }$

is said to be jointly convex if for all

${displaystyle n}$

and all

${displaystyle A_{1},A_{2}in mathbf {H} _{n}}$

with eigenvalues in

${displaystyle I}$

and all

${displaystyle B_{1},B_{2}in mathbf {H} _{n}}$

with eigenvalues in

${displaystyle J,}$

and any

${displaystyle 0leq lambda leq 1}$

the following holds

A function

${displaystyle g}$

is jointly concave if −

${displaystyle g}$

is jointly convex, i.e. the inequality above for

${displaystyle g}$

is reversed.

Trace function[edit]

Given a function

${displaystyle f:mathbb {R} to mathbb {R} ,}$

the associated trace function on

${displaystyle mathbf {H} _{n}}$

is given by

where

${displaystyle A}$

has eigenvalues

${displaystyle lambda }$

and

${displaystyle operatorname {Tr} }$

stands for a trace of the operator.

Convexity and monotonicity of the trace function[edit]

Let f: ℝ → ℝ be continuous, and let n be any integer. Then, if

${displaystyle tmapsto f(t)}$

is monotone increasing, so
is

${displaystyle Amapsto operatorname {Tr} f(A)}$

on H_n.

Likewise, if

${displaystyle tmapsto f(t)}$

is convex, so is

${displaystyle Amapsto operatorname {Tr} f(A)}$

on H_n, and
it is strictly convex if f is strictly convex.

See proof and discussion in,^[1] for example.

Löwner–Heinz theorem[edit]

For

${displaystyle -1leq pleq 0}$

, the function

${displaystyle f(t)=-t^{p}}$

is operator monotone and operator concave.

For

${displaystyle 0leq pleq 1}$

, the function

${displaystyle f(t)=t^{p}}$

is operator monotone and operator concave.

For

${displaystyle 1leq pleq 2}$

, the function

${displaystyle f(t)=t^{p}}$

is operator convex. Furthermore,

${displaystyle f(t)=log(t)}$

is operator concave and operator monotone, while

${displaystyle f(t)=tlog(t)}$

is operator convex.

The original proof of this theorem is due to K. Löwner who gave a necessary and sufficient condition for f to be operator monotone.^[5] An elementary proof of the theorem is discussed in ^[1] and a more general version of it in.^[6]

Klein’s inequality[edit]

For all Hermitian n×n matrices A and B and all differentiable convex functions
f: ℝ → ℝ with derivative f ‘ , or for all positive-definite Hermitian n×n matrices A and B, and all differentiable
convex functions f:(0,∞) → ℝ, the following inequality holds,

In either case, if f is strictly convex, equality holds if and only if A = B.
A popular choice in applications is f(t) = t log t, see below.

Proof[edit]

Let

${displaystyle C=A-B}$

so that, for

${displaystyle tin (0,1)}$

,

${displaystyle B+tC=(1-t)B+tA}$

,

varies from

${displaystyle B}$

to

${displaystyle A}$

.

Define

${displaystyle F(t)=operatorname {Tr} [f(B+tC)]}$

.

By convexity and monotonicity of trace functions,

${displaystyle F(t)}$

is convex, and so for all

${displaystyle tin (0,1)}$

,

${displaystyle F(0)+t(F(1)-F(0))geq F(t)}$

,

which is,

${displaystyle F(1)-F(0)geq {frac {F(t)-F(0)}{t}}}$

,

and, in fact, the right hand side is monotone decreasing in

${displaystyle t}$

.

Taking the limit

${displaystyle tto 0}$

yields,

${displaystyle F(1)-F(0)geq F'(0)}$

,

which with rearrangement and substitution is Klein’s inequality:

${displaystyle mathrm {tr} [f(A)-f(B)-(A-B)f'(B)]geq 0}$

Note that if

${displaystyle f(t)}$

is strictly convex and

${displaystyle Cneq 0}$

, then

${displaystyle F(t)}$

is strictly convex. The final assertion follows from this and the fact that

${displaystyle {tfrac {F(t)-F(0)}{t}}}$

is monotone decreasing in

${displaystyle t}$

.

Golden–Thompson inequality[edit]

In 1965, S. Golden ^[7] and C.J. Thompson ^[8] independently discovered that

For any matrices

${displaystyle A,Bin mathbf {H} _{n}}$

,

${displaystyle operatorname {Tr} e^{A+B}leq operatorname {Tr} e^{A}e^{B}.}$

This inequality can be generalized for three operators:^[9] for non-negative operators

${displaystyle A,B,Cin mathbf {H} _{n}^{+}}$

,

${displaystyle operatorname {Tr} e^{ln A-ln B+ln C}leq int _{0}^{infty }operatorname {Tr} A(B+t)^{-1}C(B+t)^{-1},operatorname {d} t.}$

Peierls–Bogoliubov inequality[edit]

Let

${displaystyle R,Fin mathbf {H} _{n}}$

be such that Tr e^R = 1.
Defining g = Tr Fe^R, we have

${displaystyle operatorname {Tr} e^{F}e^{R}geq operatorname {Tr} e^{F+R}geq e^{g}.}$

The proof of this inequality follows from the above combined with Klein’s inequality. Take f(x) = exp(x), A=R + F, and B = R + gI.^[10]

Gibbs variational principle[edit]

Let

${displaystyle H}$

be a self-adjoint operator such that

${displaystyle e^{-H}}$

is trace class. Then for any

${displaystyle gamma geq 0}$

with

${displaystyle operatorname {Tr} gamma =1,}$

${displaystyle operatorname {Tr} gamma H+operatorname {Tr} gamma ln gamma geq -ln operatorname {Tr} e^{-H},}$

with equality if and only if

${displaystyle gamma =exp(-H)/operatorname {Tr} exp(-H).}$

Lieb’s concavity theorem[edit]

The following theorem was proved by E. H. Lieb in.^[9] It proves and generalizes a conjecture of E. P. Wigner, M. M. Yanase and F. J. Dyson.^[11] Six years later other proofs were given by T. Ando ^[12] and B. Simon,^[3] and several more have been given since then.

For all

${displaystyle mtimes n}$

matrices

${displaystyle K}$

, and all

${displaystyle q}$

and

${displaystyle r}$

such that

${displaystyle 0leq qleq 1}$

and

${displaystyle 0leq rleq 1}$

, with

${displaystyle q+rleq 1}$

the real valued map on

${displaystyle mathbf {H} _{m}^{+}times mathbf {H} _{n}^{+}}$

given by

${displaystyle F(A,B,K)=operatorname {Tr} (K^{*}A^{q}KB^{r})}$

• is jointly concave in
  ${displaystyle (A,B)}$

  

• is convex in
  ${displaystyle K}$

  .

Here

${displaystyle K^{*}}$

stands for the adjoint operator of

${displaystyle K.}$

Lieb’s theorem[edit]

For a fixed Hermitian matrix

${displaystyle Lin mathbf {H} _{n}}$

, the function

${displaystyle f(A)=operatorname {Tr} exp{L+ln A}}$

is concave on

${displaystyle mathbf {H} _{n}^{++}}$

.

The theorem and proof are due to E. H. Lieb,^[9] Thm 6, where he obtains this theorem as a corollary of Lieb’s concavity Theorem.
The most direct proof is due to H. Epstein;^[13] see M.B. Ruskai papers,^[14][15] for a review of this argument.

Ando’s convexity theorem[edit]

T. Ando’s proof ^[12] of Lieb’s concavity theorem led to the following significant complement to it:

For all

${displaystyle mtimes n}$

matrices

${displaystyle K}$

, and all

${displaystyle 1leq qleq 2}$

and

${displaystyle 0leq rleq 1}$

with

${displaystyle q-rgeq 1}$

, the real valued map on

${displaystyle mathbf {H} _{m}^{++}times mathbf {H} _{n}^{++}}$

given by

${displaystyle (A,B)mapsto operatorname {Tr} (K^{*}A^{q}KB^{-r})}$

is convex.

Joint convexity of relative entropy[edit]

For two operators

${displaystyle A,Bin mathbf {H} _{n}^{++}}$

define the following map

${displaystyle R(Aparallel B):=operatorname {Tr} (Alog A)-operatorname {Tr} (Alog B).}$

For density matrices

${displaystyle rho }$

and

${displaystyle sigma }$

, the map

${displaystyle R(rho parallel sigma )=S(rho parallel sigma )}$

is the Umegaki’s quantum relative entropy.

Note that the non-negativity of

${displaystyle R(Aparallel B)}$

follows from Klein’s inequality with

${displaystyle f(t)=tlog t}$

.

Statement[edit]

The map

${displaystyle R(Aparallel B):mathbf {H} _{n}^{++}times mathbf {H} _{n}^{++}rightarrow mathbf {R} }$

is jointly convex.

Proof[edit]

For all

${displaystyle 0$

,

${displaystyle (A,B)mapsto operatorname {Tr} (B^{1-p}A^{p})}$

is jointly concave, by Lieb’s concavity theorem, and thus

${displaystyle (A,B)mapsto {frac {1}{p-1}}(operatorname {Tr} (B^{1-p}A^{p})-operatorname {Tr} A)}$

is convex. But

${displaystyle lim _{prightarrow 1}{frac {1}{p-1}}(operatorname {Tr} (B^{1-p}A^{p})-operatorname {Tr} A)=R(Aparallel B),}$

and convexity is preserved in the limit.

The proof is due to G. Lindblad.^[16]

Jensen’s operator and trace inequalities[edit]

The operator version of Jensen’s inequality is due to C. Davis.^[17]

A continuous, real function

${displaystyle f}$

on an interval

${displaystyle I}$

satisfies Jensen’s Operator Inequality if the following holds

${displaystyle fleft(sum _{k}A_{k}^{*}X_{k}A_{k}right)leq sum _{k}A_{k}^{*}f(X_{k})A_{k},}$

for operators

${displaystyle {A_{k}}_{k}}$

with

${displaystyle sum _{k}A_{k}^{*}A_{k}=1}$

and for self-adjoint operators

${displaystyle {X_{k}}_{k}}$

with spectrum on

${displaystyle I}$

.

See,^[17][18] for the proof of the following two theorems.

Jensen’s trace inequality[edit]

Let f be a continuous function defined on an interval I and let m and n be natural numbers. If f is convex, we then have the inequality

${displaystyle operatorname {Tr} {Bigl (}f{Bigl (}sum _{k=1}^{n}A_{k}^{*}X_{k}A_{k}{Bigr )}{Bigr )}leq operatorname {Tr} {Bigl (}sum _{k=1}^{n}A_{k}^{*}f(X_{k})A_{k}{Bigr )},}$

for all (X₁, … , X_n) self-adjoint m × m matrices with spectra contained in I and
all (A₁, … , A_n) of m × m matrices with

${displaystyle sum _{k=1}^{n}A_{k}^{*}A_{k}=1.}$

Conversely, if the above inequality is satisfied for some n and m, where n > 1, then f is convex.

Jensen’s operator inequality[edit]

For a continuous function

${displaystyle f}$

defined on an interval

${displaystyle I}$

the following conditions are equivalent:

• 
  ${displaystyle f}$

  is operator convex.

• For each natural number
  ${displaystyle n}$

  we have the inequality

${displaystyle f{Bigl (}sum _{k=1}^{n}A_{k}^{*}X_{k}A_{k}{Bigr )}leq sum _{k=1}^{n}A_{k}^{*}f(X_{k})A_{k},}$

for all

${displaystyle (X_{1},ldots ,X_{n})}$

bounded, self-adjoint operators on an arbitrary Hilbert space

${displaystyle {mathcal {H}}}$

with
spectra contained in

${displaystyle I}$

and all

${displaystyle (A_{1},ldots ,A_{n})}$

on

${displaystyle {mathcal {H}}}$

with

${displaystyle sum _{k=1}^{n}A_{k}^{*}A_{k}=1.}$

• 
  ${displaystyle f(V^{*}XV)leq V^{*}f(X)V}$

  for each isometry

  ${displaystyle V}$

  on an infinite-dimensional Hilbert space

  ${displaystyle {mathcal {H}}}$

  and

every self-adjoint operator

${displaystyle X}$

with spectrum in

${displaystyle I}$

.

Araki–Lieb–Thirring inequality[edit]

E. H. Lieb and W. E. Thirring proved the following inequality in ^[19] 1976: For any

${displaystyle Ageq 0,}$

${displaystyle Bgeq 0}$

and

${displaystyle rgeq 1,}$

In 1990 ^[20] H. Araki generalized the above inequality to the following one: For any

${displaystyle Ageq 0,}$

${displaystyle Bgeq 0}$

and

${displaystyle qgeq 0,}$

for

${displaystyle rgeq 1,}$

and

for

${displaystyle 0leq rleq 1.}$

There are several other inequalities close to the Lieb–Thirring inequality, such as the following:^[21] for any

${displaystyle Ageq 0,}$

${displaystyle Bgeq 0}$

and

${displaystyle alpha in [0,1],}$

and even more generally:^[22] for any

${displaystyle Ageq 0,}$

${displaystyle Bgeq 0,}$

${displaystyle rgeq 1/2}$

and

${displaystyle cgeq 0,}$

The above inequality generalizes the previous one, as can be seen by exchanging

${displaystyle A}$

by

${displaystyle B^{2}}$

and

${displaystyle B}$

by

${displaystyle A^{(1-alpha )/2}}$

with

${displaystyle alpha =2c/(2c+2)}$

and using the cyclicity of the trace, leading to

Effros’s theorem and its extension[edit]

E. Effros in ^[23] proved the following theorem.

If

${displaystyle f(x)}$

is an operator convex function, and

${displaystyle L}$

and

${displaystyle R}$

are commuting bounded linear operators, i.e. the commutator

${displaystyle [L,R]=LR-RL=0}$

, the perspective

${displaystyle g(L,R):=f(LR^{-1})R}$

is jointly convex, i.e. if

${displaystyle L=lambda L_{1}+(1-lambda )L_{2}}$

and

${displaystyle R=lambda R_{1}+(1-lambda )R_{2}}$

with

${displaystyle [L_{i},R_{i}]=0}$

(i=1,2),

${displaystyle 0leq lambda leq 1}$

,

${displaystyle g(L,R)leq lambda g(L_{1},R_{1})+(1-lambda )g(L_{2},R_{2}).}$

Ebadian et al. later extended the inequality to the case where

${displaystyle L}$

and

${displaystyle R}$

do not commute .^[24]

Von Neumann’s trace inequality and related results[edit]

Von Neumann’s trace inequality, named after its originator John von Neumann, states that for any
${displaystyle ntimes n}$

complex matrices

${displaystyle A}$

and

${displaystyle B}$

with singular values

${displaystyle alpha _{1}geq alpha _{2}geq cdots geq alpha _{n}}$

and

${displaystyle beta _{1}geq beta _{2}geq cdots geq beta _{n}}$

respectively,^[25]

$${displaystyle |operatorname {Tr} (AB)|~leq ~sum _{i=1}^{n}alpha _{i}beta _{i},,}$$

with equality if and only if

${displaystyle A}$

and

${displaystyle B}$

share singular vectors.^[26]

A simple corollary to this is the following result:^[27] For Hermitian

${displaystyle ntimes n}$

positive semi-definite complex matrices

${displaystyle A}$

and

${displaystyle B}$

where now the eigenvalues are sorted decreasingly (

${displaystyle a_{1}geq a_{2}geq cdots geq a_{n}}$

and

${displaystyle b_{1}geq b_{2}geq cdots geq b_{n},}$

respectively),

$${displaystyle sum _{i=1}^{n}a_{i}b_{n-i+1}~leq ~operatorname {Tr} (AB)~leq ~sum _{i=1}^{n}a_{i}b_{i},.}$$

See also[edit]

References[edit]

1. ^ ^{a b c} E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2010) 73–140 doi:10.1090/conm/529/10428
2. ^ R. Bhatia, Matrix Analysis, Springer, (1997).
3. ^ ^{a b} B. Simon, Trace Ideals and their Applications, Cambridge Univ. Press, (1979); Second edition. Amer. Math. Soc., Providence, RI, (2005).
4. ^ M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer, (1993).
5. ^ Löwner, Karl (1934). “Über monotone Matrixfunktionen”. Mathematische Zeitschrift (in German). Springer Science and Business Media LLC. 38 (1): 177–216. doi:10.1007/bf01170633. ISSN 0025-5874. S2CID 121439134.
6. ^ W.F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer, (1974).
7. ^ Golden, Sidney (1965-02-22). “Lower Bounds for the Helmholtz Function”. Physical Review. American Physical Society (APS). 137 (4B): B1127–B1128. Bibcode:1965PhRv..137.1127G. doi:10.1103/physrev.137.b1127. ISSN 0031-899X.
8. ^ Thompson, Colin J. (1965). “Inequality with Applications in Statistical Mechanics”. Journal of Mathematical Physics. AIP Publishing. 6 (11): 1812–1813. Bibcode:1965JMP…..6.1812T. doi:10.1063/1.1704727. ISSN 0022-2488.
9. ^ ^{a b c} Lieb, Elliott H (1973). “Convex trace functions and the Wigner-Yanase-Dyson conjecture”. Advances in Mathematics. 11 (3): 267–288. doi:10.1016/0001-8708(73)90011-x. ISSN 0001-8708.
10. ^ D. Ruelle, Statistical Mechanics: Rigorous Results, World Scient. (1969).
11. ^ Wigner, Eugene P.; Yanase, Mutsuo M. (1964). “On the Positive Semidefinite Nature of a Certain Matrix Expression”. Canadian Journal of Mathematics. Canadian Mathematical Society. 16: 397–406. doi:10.4153/cjm-1964-041-x. ISSN 0008-414X. S2CID 124032721.
12. ^ ^{a b} Ando, T. (1979). “Concavity of certain maps on positive definite matrices and applications to Hadamard products”. Linear Algebra and Its Applications. Elsevier BV. 26: 203–241. doi:10.1016/0024-3795(79)90179-4. ISSN 0024-3795.
13. ^ Epstein, H. (1973). “Remarks on two theorems of E. Lieb”. Communications in Mathematical Physics. Springer Science and Business Media LLC. 31 (4): 317–325. Bibcode:1973CMaPh..31..317E. doi:10.1007/bf01646492. ISSN 0010-3616. S2CID 120096681.
14. ^ Ruskai, Mary Beth (2002). “Inequalities for quantum entropy: A review with conditions for equality”. Journal of Mathematical Physics. AIP Publishing. 43 (9): 4358–4375. arXiv:quant-ph/0205064. Bibcode:2002JMP….43.4358R. doi:10.1063/1.1497701. ISSN 0022-2488. S2CID 3051292.
15. ^ Ruskai, Mary Beth (2007). “Another short and elementary proof of strong subadditivity of quantum entropy”. Reports on Mathematical Physics. Elsevier BV. 60 (1): 1–12. arXiv:quant-ph/0604206. Bibcode:2007RpMP…60….1R. doi:10.1016/s0034-4877(07)00019-5. ISSN 0034-4877. S2CID 1432137.
16. ^ Lindblad, Göran (1974). “Expectations and entropy inequalities for finite quantum systems”. Communications in Mathematical Physics. Springer Science and Business Media LLC. 39 (2): 111–119. Bibcode:1974CMaPh..39..111L. doi:10.1007/bf01608390. ISSN 0010-3616. S2CID 120760667.
17. ^ ^{a b} C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8, 42–44, (1957).
18. ^ Hansen, Frank; Pedersen, Gert K. (2003-06-09). “Jensen’s Operator Inequality”. Bulletin of the London Mathematical Society. 35 (4): 553–564. arXiv:math/0204049. doi:10.1112/s0024609303002200. ISSN 0024-6093. S2CID 16581168.
19. ^ E. H. Lieb, W. E. Thirring, Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities, in Studies in Mathematical Physics, edited E. Lieb, B. Simon, and A. Wightman, Princeton University Press, 269–303 (1976).
20. ^ Araki, Huzihiro (1990). “On an inequality of Lieb and Thirring”. Letters in Mathematical Physics. Springer Science and Business Media LLC. 19 (2): 167–170. Bibcode:1990LMaPh..19..167A. doi:10.1007/bf01045887. ISSN 0377-9017. S2CID 119649822.
21. ^ Z. Allen-Zhu, Y. Lee, L. Orecchia, Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver, in ACM-SIAM Symposium on Discrete Algorithms, 1824–1831 (2016).
22. ^ L. Lafleche, C. Saffirio, Strong Semiclassical Limit from Hartree and
    Hartree-Fock to Vlasov-Poisson Equation, arXiv:2003.02926 [math-ph].
23. ^ Effros, E. G. (2009-01-21). “A matrix convexity approach to some celebrated quantum inequalities”. Proceedings of the National Academy of Sciences USA. Proceedings of the National Academy of Sciences. 106 (4): 1006–1008. arXiv:0802.1234. Bibcode:2009PNAS..106.1006E. doi:10.1073/pnas.0807965106. ISSN 0027-8424. PMC 2633548. PMID 19164582.
24. ^ Ebadian, A.; Nikoufar, I.; Eshaghi Gordji, M. (2011-04-18). “Perspectives of matrix convex functions”. Proceedings of the National Academy of Sciences. Proceedings of the National Academy of Sciences USA. 108 (18): 7313–7314. Bibcode:2011PNAS..108.7313E. doi:10.1073/pnas.1102518108. ISSN 0027-8424. PMC 3088602.
25. ^ Mirsky, L. (December 1975). “A trace inequality of John von Neumann”. Monatshefte für Mathematik. 79 (4): 303–306. doi:10.1007/BF01647331. S2CID 122252038.
26. ^ Carlsson, Marcus (2021). “von Neumann’s trace inequality for Hilbert-Schmidt operators”. Expositiones Mathematicae. 39 (1): 149–157. doi:10.1016/j.exmath.2020.05.001.
27. ^ Marshall, Albert W.; Olkin, Ingram; Arnold, Barry (2011). Inequalities: Theory of Majorization and Its Applications (2nd ed.). New York: Springer. p. 340-341. ISBN 978-0-387-68276-1.