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volume 6, issue 4, article 105, 2005.

Received 24 August, 2005;

accepted 13 September, 2005.

Communicated by:B. Yang

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Journal of Inequalities in Pure and Applied Mathematics

ON THE DETERMINANTAL INEQUALITIES

SHILIN ZHAN

Department of Mathematics Hanshan Teacher’s College

Chaozhou, Guangdong, 521041, China.

EMail:shilinzhan@163.com

c

2000Victoria University ISSN (electronic): 1443-5756 247-05

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On the Determinantal Inequalities

Shilin Zhan

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J. Ineq. Pure and Appl. Math. 6(4) Art. 105, 2005

http://jipam.vu.edu.au

Abstract

In this paper, we discuss the determinantal inequalities over arbitrary complex matrices, and give some sufficient conditions for

d[A+B]^t≥d[A]^t+d[B]^t,

wheret∈Randt≥ _n². IfBis nonsingular andReλ(B⁻¹A)≥0, the sufficient and necessary condition is given for the above equality att= ²_n. The famous Minkowski inequality and many recent results about determinantal inequalities are extended.

2000 Mathematics Subject Classification:15A15, 15A57.

Key words: Minkowski inequality, Determinantal inequality, Positive definite matrix, Eigenvalue.

Research supported by the NSF of Guangdong Province (04300023) and NSF of Education Commission of Guangdong Province (Z03095).

1. Preliminaries

We use conventional notions and notations, as in [2]. Let A ∈ M_n(C), d[A]

stands for the modulus of det(A)(or|A|), wheredet(A)is the determinant of A. σ(A) is the spectrum ofA, namely the set of eigenvalues of matrixA. A matrixX ∈M_n(C)is called complex (semi-) positive definite ifRe(x^∗Ax)>0 (Re(x^∗Ax) ≥ 0) for all nonzerox ∈ Cⁿor if ¹₂(X+X^∗)is a complex (semi- )positive definite matrix (see [4, 7, 8, 2]). Throughout this paper, we denote C =B⁻¹AforA, B ∈M_n(C)andB is invertible.

The famous Minkowski inequality states:

IfA, B ∈M_n(R)are real positive definite symmetric matrices, then (1.1) |A+B|ⁿ¹ ≥ |A|ⁿ¹ +|B|ⁿ¹ .

It is a very interesting work to generalize the Minkowski inequality. Obvi- ously, (1.1) holds if A, B ∈ M_n(C)are positive definite Hermitian matrices.

Recently, (1.1) has been generalized forA, B ∈M_n(C)positive definite matrices (see [8], [9], [10], [3]).

In this paper, we discuss determinantal inequalities over arbitrary complex matrices, and give some sufficient conditions for

(1.2) d[A+B]^t≥d[A]^t+d[B]^t, wheret∈R.

If B is nonsingular and Reλ(B⁻¹A) ≥ 0, a sufficient and necessary condition has been given for equality as t = _n² in (1.2). The famous Minkowski inequality and many results about determinantal inequalities are extended.

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Forc∈C,Re(c)denotes the real part ofcand|c|denotes the modulus ofc.

Lett >0be fixed, we have

Lemma 1.1. IfA, B ∈ M_n(C)and B is invertible,σ(C) = {λ₁, λ₂, . . . , λ_n}, then inequality(1.2)is true if and only if

(1.3)

n

Y

i=1

|λ_i+ 1|^t≥

n

Y

i=1

|λ_i|^t+ 1,

with equality holding in(1.2)if and only if it holds in(1.3).

Proof. Sinced[A+B]^t=d[B]^td[C+I]^tandd[A]^t+d[B]^t=d[B]^t(1 +d[C]^t), formula (1.2) is equivalent to

(1.4) d[C+I]^t ≥1 +d[C]^t.

Noticeσ(C+I) ={λ_k+ 1 :k = 1,2, . . . , n}, d[C+I]^t =

n

Y

i=1

|λi+ 1|^t and d[C]^t=

n

Y

i=1

|λi|^t,

we obtain that formula (1.4) is equivalent to (1.3). Similarly, it is easy to see that the case of equality is true. Thus the lemma is proved.

Lemma 1.2 (see [6]). Ifx_t, y_t≥0 (t = 1,2, . . . , n), then

n

Y

t=1

(x_t+y_t)¹ⁿ ≥

n

Y

t=1

x

1 n

t +

n

Y

t=1

y

1 n

t ,

with equality if and only if there is linear dependence between(x₁, x₂, . . . , x_n) and(y1, y2, . . . , yn)orxt+yt = 0for a certain numbert.

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Lemma 1.3 (Jensen’s inequality). Ifa₁,a₂,. . . , a_mare positive numbers, then

n

X

i=1

a^s_i

!¹_s

≤

n

X

i=1

a^r_i

!¹_r

for 0< r≤s, n≥2.

Lemma 1.4. IfP₁,P₂,. . . , P_mare positive numbers andT ≥ _m¹, then

(1.5)

m

Y

k=1

(P_k+ 1)^T ≥

m

Y

k=1

P_k^T + 1,

with equality if and only ifP_k(k = 1,2, . . . , n)is constant asT = _m¹. Proof. By Lemma1.2, we have

m

Y

k=1

(P_k+ 1)^T =

" _m Y

k=1

(P_k+ 1)^m¹

#mT

≥

" _m Y

k=1

P_k^T_mT¹ + 1

#mT

.

On noting that0< _mT¹ ≤1, by Lemma1.3, we obtain

" _m Y

k=1

P_k^T_mT¹ + 1

#mT

≥

m

Y

k=1

P_k^T + 1,

and inequality (1.5) is demonstrated. By Lemma1.2, it is easy to see that equality holds if and only ifP_k(k = 1,2, . . . , n)is constant asT = _m¹.

Remark 1. Apparently, Lemma1.3is tenable fora_i ≥0 (i= 1,2, . . . , n), and Lemma1.4is tenable forP_i ≥0 (i= 1,2, . . . , n).

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2. Main Results

Theorem 2.1. Let A, B ∈ M_n(C). If B is nonsingular and Reλ_k ≥ 0 (k = 1,2, . . . , n), whereσ(C) = {λ1, λ2, . . . , λn},then fort ≥ ²_n

(2.1) d[A+B]^t≥d[A]^t+d[B]^t,

Proof. By Lemma 1.1, we need to prove inequality (1.3). Note thatReλ_k ≥ 0 (k = 1,2, . . . , n)and|λk+ 1|² ≥1 +|λk|²,

n

Y

k=1

|λ_k+ 1|^t=

n

Y

k=1

|λ_k+ 1|²

!^t₂

≥

n

Y

k=1

|λ_k|²+ 1₂^t .

Applying Lemma1.4, we can show that

n

Y

k=1

(|λ_k|²+ 1)²^t ≥

n

Y

k=1

|λ_k|^t+ 1 for t≥ 2 n,

with equality if and only if |λ_k|² (k = 1,2, . . . , n) is constant ast = _n². The above two inequalities imply formula (1.3).

Whent= 1, we have

Corollary 2.2. Let A, B ∈ M_n(C) (n ≥ 2). IfB is invertible andReλ_k ≥ 0 (k = 1,2, . . . , n), whereσ(C) ={λ1, λ2, . . . , λn}, then

(2.2) d[A+B]≥d[A] +d[B].

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Corollary 2.3. LetAbe ann-by-ncomplex positive definite matrix, andB be ann-by-npositive definite Hermitian matrix(n≥2). Then fort≥ _n²

(2.3) d[A+B]^t≥d[A]^t+ [det(B)]^t.

Proof. ObservingC =B⁻¹Ais similar toB⁻¹²AB⁻¹² andReλ(B⁻¹²AB⁻¹²)>

0, where λ(B⁻¹²AB⁻¹²) is an arbitrary eigenvalue of B⁻¹²AB⁻¹². Therefore, Reλ_k ≥0andσ(C) ={λ₁, λ₂, . . . , λ_n}. Hence, Theorem2.1yields Corollary 2.3.

Whent= _n², inequality (2.3) gives Theorem 4 of [3]. Whent= 1, inequality (2.3) gives Theorem 1 of [3]. To merit attention, Theorem 2 in [8] proves that ifAis real positive definite andB is real positive definite symmetric, then (2.3) holds fort= ¹_n. It is untenable for example: A=

1 1

−1 1

, B =

1 0 0 1

. Corollary2.7and Corollary2.8in this paper have been given correction.

Theorem 2.4. LetA, B ∈ M_n(C). IfB is nonsingular, andReλ_k ≥ 0 (k = 1,2, . . . , n), where σ(C) = {λ₁, λ₂, . . . , λ_n}, thenneigenvalues of Care pure imaginary complex numbers with the same modulus if and only if

(2.4) d[A+B]ⁿ² =d[A]²ⁿ +d[B]ⁿ², Proof. Ifneigenvalues ofCare±id(i=√

−1, d > o, d∈R), then

n

Y

i=1

|λ_i+ 1|²ⁿ =

n

Y

i=1

1 +d²¹_n

= 1 +d² =

n

Y

i=1

|λ_i|²ⁿ + 1.

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Hence equality (2.4) holds by Lemma1.1.

Conversely, suppose (2.4) holds, then

n

Y

i=1

|λ_i+ 1|²ⁿ =

n

Y

i=1

|λ_i|²ⁿ + 1.

So n

Y

i=1

(1 + 2 Reλ_i+|λ_i|²)¹ⁿ =

n

Y

i=1

(|λ_i|²)¹ⁿ + 1.

Obviously,Reλ_k= 0 (k = 1,2, . . . , n), otherwise

n

Y

i=1

(1 + 2 Reλi+|λi|²)¹ⁿ >

n

Y

i=1

1 +|λi|²¹_n

≥

n

Y

i=1

(|λi|²)¹ⁿ + 1,

with illogicality. Therefore

n

Y

i=1

1 + (Imλ_i)²¹_n

=

n

Y

i=1

(Imλ_i)²¹_n + 1.

By Lemma1.2we obtain(Imλ_k)² =d² andλ_k =±id(k = 1,2, . . . , n). This completes the proof.

Corollary 2.5. If A,B ∈ M_n(C) with B is nonsingular and C = B⁻¹A is skew–Hermitian, then formula(2.4)holds if and only ifA=idBU EU^∗, where i² =−1,d > 0, U is a unitary matrix,E = diag(e₁, e₂, . . . , e_n)withe_i =±1, i= 1,2, . . . , n.

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Proof. SinceC is skew–Hermitian and its real parts ofn eigenvalues are zero, then Theorem2.4implies that (2.4) holds if and only if

C =B⁻¹A=U diag(±id,±id, . . . ,±id)U^∗,

where σ(C) = {±id,±id, . . . ,±id}, d > 0 and U is unitary. Hence A = idBU EU^∗, wherei² =−1,d >0,Uis a unitary matrix,E = diag(e₁, e₂, . . . , e_n) ande_i =±1,i= 1,2, . . . , n.

Theorem 2.6. Suppose A, B ∈ M_n(C) with B nonsingular and Reλ_k ≥ 0 (k = 1,2, . . . , n), whereσ(C) = {λ₁, λ₂, . . . , λ_n}. If the number of the real eigenvalues ofCisr, and the non-real eigenvalues ofCare pair wise conjugate, then inequality(1.2)holds fort ≥ _n+r² .

Proof. By Lemma 1.1, we need to prove (1.3) for t ≥ _n+r² . Without loss of generality, suppose λj ≥ 0 (j = 1,2, . . . , r)are the real eigenvalues ofC and λ_k, λ_k (k = r+ 1, r+ 2, . . . , r+s)ares pairs of non-real eigenvalues ofC, wheren=r+ 2s. Then the right-hand side of (1.3) becomes

(2.5)

r

Y

i=1

λ^t_i

r+s

Y

j=r+1

|λj|²_t + 1,

and the left-hand side of (1.3) is

(2.6)

r

Y

i=1

(λ_i+ 1)^t

r+s

Y

j=r+1

|1 +λ_j|²_t .

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GivenReλ_k ≥0 (k= 1,2, . . . , r+s), so|1 +λ_j|² ≥1 +|λ_j|², then

(2.7)

r

Y

i=1

(1 +λ_i)^t

r+s

Y

j=r+1

|1 +λ_j|²_t

≥

r

Y

i=1

(1 +λ_i)^t

r+s

Y

j=r+1

1 +|λ_j|²_t .

By Lemma1.2and (2.7), we obtain that

r

Y

i=1

(λ_i+ 1)^t

r+s

Y

j=r+1

|1 +λ_j|²_t

≥

r

Y

i=1

λ^t_i

r+s

Y

j=r+1

|λ_j|²_t + 1,

for 1

r+s = 2 n+r. This completes the proof.

In the following, we present some generalizations of the Minkowski inequality. By Theorem2.6, it is easy to show:

Corollary 2.7. LetA, B ∈M_n(C). IfBis nonsingular andneigenvalues ofC are positive numbers, then fort≥ ¹_n

(2.8) d[A+B]ⁿ¹ ≥d[A]ⁿ¹ +d[B]ⁿ¹.

IfAis ann-by-ncomplex positive definite matrix andB is ann-by-n positive definite Hermitian matrix, withneigenvalues ofCbeing real numbers, then σ(C) =σ(B¹²CB⁻¹²), andB¹²CB⁻¹² =B⁻¹²AB⁻¹² is positive definite, so any eigenvalue of C has a positive real part. Thusn eigenvalues ofC are positive numbers. By Corollary2.7we have

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Corollary 2.8. SupposeA, B ∈M_n(C),whereAis a complex positive definite matrix and B is a positive definite Hermitian matrix. Ifneigenvalues of Care real numbers, then inequality(2.8)holds fort≥ ¹_n.

Corollary 2.9 (Minkowski inequality). Suppose A, B ∈ M_n(C)are positive definite Hermitian matrices, then inequality(1.1)holds.

Proof. Note that C = B⁻¹Ais similar to a real diagonal matrix, and its eigenvalues are real numbers, using Corollary 2.8 and letting t = 1, the proof is completed.

Corollary 2.10. SupposeA, B ∈M_n(C),whereAis a complex positive definite matrix andBis a positive definite Hermitian matrix. If the non-real eigenvalues of C are m pairs conjugate complex numbers, then inequality(1.2) holds for t ≥ _n−m¹ .

Proof. ObviouslyReλ_k ≥0 (k= 1,2, . . . , n), whereσ(C) ={λ₁, λ₂, . . . , λ_n}.

Applying Theorem2.6completes the proof.

LetA =H +K ∈ M_n(C), whereH = ¹₂(A+A^∗), andK = ¹₂(A−A^∗), then we have

Theorem 2.11. LetA=H+K be ann-by-ncomplex positive definite matrix, then fort≥ _n²

(2.9) d[A]^t≥d[H]^t+d[K]^t,

with equality if and only ifK =idHQ^∗EQast = _n², wherei² =−1,d >0,Q is a unitary matrix,E = diag(e₁, e₂, . . . , e_n)withe_i =±1,i= 1,2, . . . , n.

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Proof. SinceH⁻¹²KH⁻¹² is a skew-Hermitian matrix and is similar toH⁻¹K, Reλ(H⁻¹K) = Reλ(H⁻¹²KH⁻¹²) = 0. By Theorem 2.1 and Corollary 2.5, we get the desired result.

Lett = 1, we have the following interesting result.

Corollary 2.12. IfA = H +K is ann-by-n complex positive definite matrix (n ≥2), then

(2.10) d[A]≥d[H] +d[K].

Corollary 2.13 (Ostrowski-Taussky Inequality). IfA=H+K is ann-by-n positive definite matrix(n ≥2), thendetH ≤ d[A]with equality if and only if Ais Hermitian.

Theorem 2.14. LetA, B be twon-by-ncomplex positive definite matrices, and n eigenvalues of B be real numbers. Suppose A, B are simultaneously upper triangularizable, namely, there exists a nonsingular matrixP, such thatP⁻¹AP andP⁻¹BP are upper triangular matrices, then inequality(1.2)holds for any t ≥ _n².

Proof. IfP⁻¹AP andP⁻¹BP are upper triangular matrices, then P⁻¹B⁻¹AP = (P⁻¹BP)⁻¹(P⁻¹AP)

is an upper triangular matrix, with the product of the eigenvalues of B⁻¹ and A on its diagonal. We denote the eigenvalue of X by λ(X). Notice that positive definiteness ofA andB⁻¹, Reλ(A)andλ(B⁻¹) are positive numbers by hypothesis, it is easy to see thatReλ(B⁻¹A)≥0. By Theorem2.1, we get the desired result.

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Corollary 2.15. LetA, Bbe twon-by-ncomplex positive definite matrices, and all the eigenvalues ofBbe real numbers. Ifr([A, B])≤1, then inequality(1.2) holds fort≥ _n², where[A, B] =AB−BA,r([A, B])is the rank of[A, B].

Proof. It is easy to see that B⁻¹ is a complex positive definite matrix and n eigenvalues of B⁻¹ are real numbers. By the hypothesis and r[B⁻¹, A] = r[A, B], we haver([B⁻¹, A])≤ 1. By the Laffey-Choi Theorem (see [5], [1]), there exists a non-singular matrixP, such thatP⁻¹AP andP⁻¹BP are upper triangular matrices. The result holds by Theorem2.14.

Corollary 2.16. Let A, B be two n-by-n complex positive definite matrices (n ≥ 2). SupposeAB = BA andn eigenvalues of B are real numbers, then inequality(1.2)holds fort≥ _n².

Proof. Follows from Corollary2.15and the fact thatr([A, B]) = 0.

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[4] C.R. JOHNSON, Positive definite matrices, Amer. Math. Monthly, 77 (1970), 259–264.

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Sinica, 33(4) (1990), 462–471.

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[10] HUI-PENG YUAN, Minkowski inequality of complex positive definite matrix, Journal of Mathematical Research and Exposition, 21(3) (2001), 464–468.