On Some Matrix Trace Inequalities

Volume 2010, Article ID 201486,8pages doi:10.1155/2010/201486

Research Article

On Some Matrix Trace Inequalities

Z ¨ubeyde Uluk ¨ok and Ramazan T ¨urkmen

Department of Mathematics, Science Faculty, Selc¸uk University, 42003 Konya, Turkey

Correspondence should be addressed to Z ¨ubeyde Uluk ¨ok,[email protected]

Received 23 December 2009; Revised 4 March 2010; Accepted 14 March 2010

Academic Editor: Martin Bohner

Copyrightq2010 Z. Uluk ¨ok and R. T ¨urkmen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We first present an inequality for the Frobenius norm of the Hadamard product of two any square matrices and positive semidefinite matrices. Then, we obtain a trace inequality for products of two positive semidefinite block matrices by using 2×2 block matrices.

1. Introduction and Preliminaries

LetMm,n denote the space ofm×ncomplex matrices and writeMn ≡ Mn,n. The identity matrix inMn is denotedIn. As usual,A∗ AT denotes the conjugate transpose of matrix A. A matrixA ∈ Mn is Hermitian ifA∗ A. A Hermitian matrixAis said to be positive semidefinite or nonnegative definite, written asA≥0, if

x∗_{Ax≥0, ∀x∈C}n_{. 1.1}

Ais further called positive definite, symbolizedA >0, if the strict inequality in1.1holds for all nonzerox∈Cn. An equivalent condition forA∈Mnto be positive definite is thatAis Hermitian and all eigenvalues ofAare positive real numbers. Given a positive semidefinite matrixAandp >0,Apdenotes the unique positive semidefinitepth power ofA.

Let A and B be two Hermitian matrices of the same size. If A − B is positive semidefinite, we write

A≥B or B≤A. 1.2

the sum of its main diagonal entries, or, equivalently, the sum of its eigenvaluesis denoted by trA.

LetAbe anym×nmatrix. The FrobeniusEuclideannorm of matrixAis

A_F ⎡ ⎣m

i1 n

j1 _aij2

⎤ ⎦

1/2

. 1.3

It is also equal to the square root of the matrix trace ofAA∗,that is,

A_F trAA∗. 1.4

A norm · onMm,n is called unitarily invariantUAV Afor allA ∈ Mm,nand all unitaryU∈Mm, V ∈Mn.

Given two real vectorsx x1, . . . , x_nandy y1, . . . , y_nin decreasing order, we say thatxis weakly log majorized byy, denotedx≺wlogy, ifΠk_i1xi ≤Πki1yi, k 1,2, . . . , n, and we say thatxis weakly majorized byy, denotedx≺wy, if k_i1xi≤ k_i1yi, k1,2, . . . , n. We sayxis majorized byydenoted byx≺y, if

x≺wy, n

i1 xi

n

i1

yi. 1.5

As is well known,x≺wlogyyieldsx≺wysee, e.g.,1, pages 17–19. LetAbe a square complex matrix partitioned as

A

A11 A12 A21 A22

, 1.6

whereA11is a square submatrix ofA. IfA11is nonsingular, we call

A11A22−A21A−1

11A12 1.7

the Schur complement ofA11 inAsee, e.g.,2, page 175. IfAis a positive definite matrix, thenA11is nonsingular and

A22≥A11 ≥0. 1.8

Recently, Yang 3 proved two matrix trace inequalities for positive semidefinite matricesA∈MnandB∈Mn,

0≤trAB2n≤trA2trA2n−1trB2n,

0≤trAB2n1 ≤trAtrBtrA2ntrB2n,

1.9

Also, authors in 4 proved the matrix trace inequality for positive semidefinite matricesAandB,

trABm≤trA2mtrB2m1/2, 1.10

wheremis a positive integer.

Furthermore, one of the results given in5is

ndetA·detBm/n≤trAmBm 1.11

forAandBpositive definite matrices, wheremis any positive integer.

2. Lemmas

Lemma 2.1see, e.g.,6. For anyAandB∈M_n, σA◦B≺_wσA◦σB.

Lemma 2.2see, e.g.,7. LetA, B∈Mm,n,then

t

i1 δi

AB2m_≤t i1

λiA∗ABB∗m

≤t

i1

λiA∗AmBB∗m, 1≤t≤n, m∈N.

2.1

Lemma 2.3Cauchy-Schwarz inequality. Leta1, a2, . . . , an andb1, b2, . . . , bn be real numbers. Then,

_n

i1 aibi

2

≤

_n

i1 a2

i

_n

i1 b2

i

, ∀ai, bi∈R. 2.2

Lemma 2.4see, e.g.,8, page 269. IfAandBare poitive semidefinite matrices, then,

0≤trAB≤trAtrB. 2.3

Lemma 2.5see, e.g.,9, page 177. LetAandBaren×nmatrices. Then,

k

i1

siAB≤k

i1

siAsiB 1≤k≤n. 2.4

Lemma 2.6see, e.g.,10. LetFandGare positive semidefinite matrices. Then,

t

i1 λm

i FG≤ t

i1

λiFm_Gm_{, 1≤t≤n, 2.5}

3. Main Results

Horn and Mathias11show that for any unitarily invariant norm · onM_n

A∗_B2_{≤ A}∗_AB∗_{B ∀A, B∈M}

m,n,

A◦B2_{≤ A}∗_AB∗_{B ∀A, B∈M}

n.

3.1

Also, the authors in12show that for positive semidefinite matrixAL X X∗_M

, whereX ∈ Mm,n

_|X|p2

≤ Lp_Mp _3.2

for allp >0 and all unitarily invariant norms · .

By the following theorem, we present an inequality for Frobenius norm of the power of Hadamard product of two matrices.

Theorem 3.1. LetAandBben-square complex matrices. Then

_A◦Bm2

F ≤A∗AmFB∗BmF, 3.3

wheremis a positive integer. In particular, ifAandBare positive semidefinite matrices, then

_A◦Bm2

F ≤A2m_FB2m_F. 3.4

Proof. From definition of Frobenius norm, we write

_A◦Bm2 F tr

A◦BmA◦Bm∗. 3.5

Also, for anyAandB, it follows thatsee, e.g.,13

AA∗_◦BB∗ _A◦B

A∗_◦B∗ _I

≥0, 3.6

A◦BA◦B∗≤AA∗_◦BB∗_{. 3.7}

Since|trA2m| ≤trAmA∗m≤trAA∗mforA∈Mnand from inequality3.7, we write

_A◦Bm2

F trA◦BmA◦Bm∗

≤trA◦BA◦B∗m

≤trAA∗◦BB∗m.

FromLemma 2.1and Cauchy-Schwarz inequality, we write

trAm◦Bm n

i1

λiAm◦Bm≤ n

i1

λiAmλiBm

≤

_n

i1 λ2

iAm n

i1 λ2

iBm 1/2

trA2mtrB2m1/2.

3.9

By combining inequalities3.7,3.8, and3.9, we arrive at

trAA∗◦BB∗m≤trAA∗AA∗mtrBB∗BB∗m1/2

≤trAA∗AA∗mtrBB∗BB∗m1/2

trAA∗2m1/2trBB∗2m1/2

A∗_Am

FB∗BmF.

3.10

Thus, the proof is completed. LetAandBbe positive semidefinite matrices. Then

_A◦Bm2

F ≤A2m_FB2m_F, 3.11

wherem >0.

Theorem 3.2. Let A_i ∈ M_n i 1,2, . . . , kbe positive semidefinite matrices. For positive real numberss, m, t

_k

i1

A_ist/2m2 F

2

≤

_k

i1 _Asm

i 2F

_k

i1 _Atm

i 2F

. 3.12

Proof. Let A ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

AS/2₁ 0 · · · 0 0 As/2₂ · · · 0

..

. ... . .. ...

0 0 · · · As/2_k ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , B ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

At/2₁ 0 · · · 0 0 At/2₂ · · · 0

..

. ... . .. ...

0 0 · · · At/2_k ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

We know thatA, B≥0, then by using the definition of Frobenius norm, we write

_A◦Bm2 F

k

i1

A_ist/2m2 F ,

A2m

F ! ! "k

i1 _Asm

i 2F, B2m_F ! ! "k

i1 _Atm

i 2F.

3.14

Thus, by usingTheorem 3.1, the desired is obtained.

Now, we give a trace inequality for positive semidefinite block matrices.

Theorem 3.3. Let

A

A11 A12 A21 A22

≥0, B

B11 B12 B21 B22

≥0, 3.15

then,

tr#A22 1/2B1/2₁₁ $2m

tr #

A1/2₂₂ B11 1/2 $2m

≤trABm≤trAmBm, 3.16

wheremis an integer.

Proof. Let

M

X 0 Y Z

3.17

withZA1/2₂₂ , Y A₂₂−1/2A21, X A11−A12A−1

22A211/2. ThenAM∗Msee, e.g.,14. Let

K

X 0 Y Z

3.18

withZ B22−B21B−₁₁1B121/2,Y B21B₁₁−1/2,X B1/2₁₁ . ThenB KK∗ see, e.g.,14. We know that

Mk

Xk ₀

∗ Zk

,

M·K ⎡ ⎣

A11−A12A−1 22A21

1/2_B1/2

11 0

A−₂₂1/2A21B1/2₁₁ A1/2₂₂ B21B₁₁−1/2 A1/2₂₂ B22−B21B−1 11B12

M·K2m ⎡ ⎢ ⎣

&

A11−A12A−1 22A21

1/2_B1/2 11

'2m

0

∗ &A1/2₂₂ B22−B21B−1 11B12

1/2'2m ⎤ ⎥ ⎦.

3.19

By usingLemma 2.2, it follows that

trMK2m≤ n

i1

siMK2m≤n

i1

siMK2m

n

i1

s2 iMK

_m

n

i1

λiM∗_MKK∗m

n

i1

λiABm n

i1

trABm≤ n

i1

λiM∗MmKK∗m

n

i1

λiAmBm n

i1

trAmBm.

3.20

Therefore, we get

trMK2mtr#)A11−A12A−₂₂1A211/2 *

B1/2₁₁ $2m

tr #

A1/2₂₂ B22−B21B−1 11B12

_1/2$2m

≤trABm≤trAmBm.

3.21

As result, we write

tr#A22 1/2B1/2₁₁ $2m

tr #

A1/2₂₂ B11 1/2 $2m

≤trABm≤trAmBm. 3.22

Example 3.4. Let

A

4 1

1 1

>0, B

5 2

2 1

>0. 3.23

Then trAB25,detA3,detB1.From inequality1.11, form1,we get

Also, form1, since trA22+1/2B₁₁1/2215 and trA1/2₂₂ B11+1/220.2, we get

tr )

+ A221/2B₁₁1/2

*2 tr

)

A1/2₂₂ B11+1/2 *2

15.2. 3.25

Thus, according to this example from3.24and3.25, we get

ndetAdetB1/n≤tr )

+ A221/2B₁₁1/2

*2 tr

)

A1/2₂₂ B11+1/2 *2

≤trAB. 3.26

Acknowledgment

This study was supported by the Coordinatorship of Selc¸uk University’s Scientific Research ProjectsBAP.

References

1 X. Zhan,Matrix Inequalities, vol. 1790 ofLecture Notes in Mathematics, Springer, Berlin, Germany, 2002.

2 F. Zhang,Matrix Theory: Basic Results and Techniques, Universitext, Springer, New York, NY, USA, 1999.

3 X. Yang, “A matrix trace inequality,”Journal of Mathematical Analysis and Applications, vol. 250, no. 1, pp. 372–374, 2000.

4 X. M. Yang, X. Q. Yang, and K. L. Teo, “A matrix trace inequality,”Journal of Mathematical Analysis and Applications, vol. 263, no. 1, pp. 327–331, 2001.

5 F. M. Dannan, “Matrix and operator inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 2, no. 3, article 34, 7 pages, 2001.

6 F. Z. Zhang, “Another proof of a singular value inequality concerning Hadamard products of matrices,”Linear and Multilinear Algebra, vol. 22, no. 4, pp. 307–311, 1988.

7 Z. P. Yang and X. X. Feng, “A note on the trace inequality for products of Hermitian matrix power,” Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 5, article 78, 12 pages, 2002.

8 E. H. Lieb and W. Thirring, Studies in Mathematical Physics, Essays in Honor of Valentine Bartmann, Princeton University Press, Princeton, NJ, USA, 1976.

9 R. A. Horn and C. R. Johnson,Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.

10 B. Y. Wang and M. P. Gong, “Some eigenvalue inequalities for positive semidefinite matrix power products,”Linear Algebra and Its Applications, vol. 184, pp. 249–260, 1993.

11 R. A. Horn and R. Mathias, “An analog of the Cauchy-Schwarz inequality for Hadamard products and unitarily invariant norms,”SIAM Journal on Matrix Analysis and Applications, vol. 11, no. 4, pp. 481–498, 1990.

12 R. A. Horn and R. Mathias, “Cauchy-Schwarz inequalities associated with positive semidefinite matrices,”Linear Algebra and Its Applications, vol. 142, pp. 63–82, 1990.

13 F. Zhang, “Schur complements and matrix inequalities in the L ¨owner ordering,”Linear Algebra and Its Applications, vol. 321, no. 1–3, pp. 399–410, 2000.