Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices

Yan Hong; Feng Qi

doi:10.1515/math-2021-0061

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Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices

Yan Hong and Feng Qi

Published/Copyright: July 9, 2021

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$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, the authors extend determinantal inequalities of the Hua-Marcus-Zhang type for positive definite matrices to the corresponding ones for quaternion matrices.

Keywords: determinantal inequality; matrix of quaternions; eigenvalue; singular value; positive definite matrix; Hua-Marcus-Zhang type

MSC 2010: 15B33; 11R52; 15A42; 15A45; 16H05; 20G20

1 Introduction and motivations

In 1955, Hua [1,2] established an elegant determinantal inequality as follows.

Theorem 1.1

[1,2, Theorems 1 and 2] Let A , B ∈ C n × n and A ∗ , B ∗ be conjugate matrices of A , B . If I n − A ∗ A and I n − B ∗ B are positive definite, then

(1.1) det ( I n − A ∗ A ) det ( I n − B ∗ B ) + ∣ det ( A − B ) ∣ 2 ≤ ∣ det ( I n − A ∗ B ) ∣ 2 ,

where I n denotes identity matrix of n order. In particular,

det ( I n − A ∗ A ) det ( I n − B ∗ B ) ≤ ∣ det ( I n − A ∗ B ) ∣ 2 .

For A ∈ C n × n , we denote its eigenvalues and singular values by

λ 1 ( A ) , λ 2 ( A ) , … , λ n ( A ) and σ 1 ( A ) , σ 2 ( A ) , … , σ n ( A ) ,

respectively. If all eigenvalues of A ∈ C n × n are real, we assume

(1.2) λ 1 ( A ) ≥ λ 2 ( A ) ≥ ⋯ ≥ λ n ( A ) ,

while we always write

(1.3) σ 1 ( A ) ≥ σ 2 ( A ) ≥ ⋯ ≥ σ n ( A ) .

In 1958, Marcus [3] extended the determinantal inequality (1.1) to the following family of inequalities for eigenvalues of matrices.

Theorem 1.2

[3, Theorem] Let A , B ∈ C n × n . If I n − A ∗ A and I n − B ∗ B are positive definite, then inequality

(1.4) ∏ j = 1 k σ n − j + 1 2 ( I n − A ∗ B ) ≥ ∏ j = 1 k ( 1 − λ j ( A ∗ A ) ) ( 1 − λ j ( B ∗ B ) ) ,

is valid for 1 ≤ k ≤ n .

In 2011, Zhang [4] found a reversion of the determinantal inequality (1.1) below.

Theorem 1.3

[4, Theorem 7.18] If A , B ∈ C n × n , then

(1.5) ∣ det ( I n − A ∗ B ) ∣ 2 ≤ det ( I n + A ∗ A ) det ( I n + B ∗ B ) − ∣ det ( A + B ) ∣ 2 .

In the past few decades, many kinds of determinantal inequalities of the Hua-Marcus-Zhang type were presented in [3,4, 5,6,7, 8,9] and closely related references therein. In recent years, quaternions have been widely used in system control, signal and color image processing, geometric rotation, quantum mechanics, algebra, analysis, and other fields. For details, please refer to previous studies [10,11, 12,13,14, 15,16].

In this paper, we adopt notations for quaternion matrices in our own paper [17]. Hamilton [18] introduced the concept of real quaternions by

H = { x = a + b i + c j + d k : a , b , c , d ∈ R and i , j , k ∉ R } ,

where there exist the following operations:

i ¯ = − i , j ¯ = − j , k ¯ = − k , i 2 = k 2 = j 2 = − 1 , ij = − ji = k , jk = − kj = i , ki = − ik = j .

Let H n × m be the set of n × m quaternion matrices and let H n ∗ be the set of self-conjugate n × n quaternion matrices. Xie [19,20, 21,22] studied a similarity standard form of the body matrices, the eigenvalue theory of centralizable matrices, and the determinant theory of self-conjugate quaternion matrices. Zhuang [23] established the Cauchy interleaving theorem of eigenvalues of self-conjugate quaternion matrices. Cao [24] investigated inequalities between eigenvalues and singular values of products of two quaternion matrices. The papers [17,25,26, 27,28,29, 30,31] are also related to the topic of this paper. The research of quaternion matrices has been being continuously active.

Definition 1.1

[32] Let A ∈ H n × n . If there exists λ ∈ H and x ∈ H n ⧹ { 0 } such that

A x = λ x ( A x = x λ ) ,

then we say that λ and x are the left (right) eigenvalue and the corresponding left (right) eigenvector of A , respectively.

Since the quaternions do not commute, it is mandatory to treat the linear systems A x = λ x and A x = x λ separately. Rewrite the first one as ( λ I − A ) x = 0 . In the complex case, the fact that det ( λ I − A ) = 0 has a solution guarantees an eigenvalue for A . The existence of eigenvalue of a quaternion matrix is much more complicated, see [32]. It is known [33,34] that the right spectrum is always nonempty. In [35], Wood used a topological method to confirm that the left eigenvalue always exists.

Over a long period of time, there have been various kinds of definitions of determinant over the quaternion, see [19,22,36, 37,38]. Since eigenvalues of self-conjugate quaternion matrices are real numbers, see Lemma 2.1, we use the definition of determinants in [21,22] to investigate our problems in this paper.

In what follows, we assume that the eigenvalues λ j ( A ) of A ∈ H n ∗ and the singular values σ j ( A ) of A ∈ H n × n are expressed in the orders (1.2) and (1.3), respectively. In this paper, we will extend inequalities (1.1), (1.4), and (1.5) to the corresponding ones for quaternion matrices.

2 Lemmas

In order to extend inequalities (1.1), (1.4), and (1.5) to the corresponding ones for quaternion matrices, we need the following lemmas.

Lemma 2.1

[17, Proposition 2.1], [21, Theorem 13], and [22, Theorem 3] For A ∈ H n ∗ , there exists a generalized unitary matrix U ∈ H n × n (that is, U ∗ U = U U ∗ = I n ) such that

U ∗ A U = diag ( λ 1 ( A ) , λ 2 ( A ) , … , λ n ( A ) ) .

Furthermore, if A is positive definite, then the least eigenvalue λ n ( A ) is positive.

Lemma 2.2

[17, Proposition 2.2] and [24, Theorems 2 and 3] Suppose A , B ∈ H n ∗ and B is positive definite. Then

there exists a nonsingular matrix P = ( p 1 , p 2 , … , p n ) ∈ H n × n such that
P ∗ B P = I n and P ∗ A P = diag ( λ 1 ( A ) , λ 2 ( A ) , … , λ n ( A ) ) ;
the products A B and B A are both centralizable matrices and both of them are similar to a real diagonal matrix.

Lemma 2.3

Let A , B ∈ H n ∗ . If A , B are both positive semidefinite, then

(2.1) det ( A + B ) ≥ det ( A ) + det ( B ) .

Proof

When det B = 0 , inequality (2.1) holds trivially.

When det B ≠ 0 , by Lemma 2.2, there exists a nonsingular matrix P ∈ H n × n such that

P − 1 ( A + B ) P = ( P ∗ P ) − 1 ( D + I n ) ,

where D = diag ( λ 1 ( A + B ) , λ 2 ( A + B ) , … , λ n ( A + B ) ) . Since

det ( A ) = [ det ( P ∗ P ) ] − 1 det ( D ) , det ( B ) = [ det ( P ∗ P ) ] − 1 , det ( D + I n ) ≥ det ( D ) + 1 ,

using Lemma 2.2, it follows that

det ( A + B ) = [ det ( P ∗ P ) ] − 1 det ( D + I n ) ≥ det ( A ) + det ( B ) .

The proof of Lemma 2.3 is complete.□

Lemma 2.4

[23, Theorem 1] Let A ∈ H n ∗ be a matrix partitioned as

A = A k 1 A k 2 A k 3 A k 4 , A k 1 ∈ H k ∗ , 1 ≤ k ≤ n .

Then

λ ℓ ( A ) ≥ λ n − k + ℓ ( A k 1 ) ≥ λ n − k + ℓ ( A )

for 1 ≤ k , ℓ ≤ n . In particular, if A is positive semidefinite, then

(2.2) ∏ ℓ = 1 k λ ℓ ( A ) ≥ det ( A k 1 ) ≥ ∏ ℓ = 1 k λ n − ℓ + 1 ( A )

for 1 ≤ k ≤ n .

Lemma 2.5

Let A , B ∈ H n × n . For 1 ≤ k ≤ n , we have

(2.3) ∏ ℓ = 1 k σ ℓ ( A ) σ ℓ ( B ) ≥ ∏ ℓ = 1 k σ ℓ ( A B ) ≥ ∏ ℓ = 1 k σ ℓ ( A ) σ n − ℓ + 1 ( B )

(2.4) ≥ ∏ ℓ = 1 k σ n − ℓ + 1 ( A B ) ≥ ∏ ℓ = 1 k σ n − ℓ + 1 ( A ) σ n − ℓ + 1 ( B ) .

Proof

The first inequality (2.3) in Lemma 2.5 comes from [39, Theorem 3].

We assume that A and B are nonsingular, otherwise we use the continuity of A A ∗ + ε I n and B B ∗ + ε I n in ε > 0 . By inequality (2.3), we obtain

∏ ℓ = 1 k σ n − ℓ + 1 ( A − 1 ) σ ℓ ( B − 1 ) ≤ ∏ ℓ = 1 k σ ℓ ( ( A B ) − 1 ) ≤ ∏ ℓ = 1 k σ ℓ ( A − 1 ) σ ℓ ( B − 1 ) .

Since σ ℓ ( A − 1 ) = σ n − ℓ + 1 − 1 ( A ) for 1 ≤ ℓ ≤ n , we readily see that inequality (2.4) holds. The proof of Lemma 2.5 is complete.□

Lemma 2.6

Let A , B ∈ H n × n . Then

(2.5) I n + A A ∗ = ( A + B ) ( I n + B ∗ B ) − 1 ( A + B ) ∗ + ( I n − A B ∗ ) ( I n + B B ∗ ) − 1 ( I n − A B ∗ ) ∗ .

If I n − A A ∗ and I n − B B ∗ are nonsingular, then

(2.6) ( I n − B ∗ A ) ( I n − A ∗ A ) − 1 ( I n − A ∗ B ) = ( I n − B ∗ B ) + ( A − B ) ∗ ( I n − A A ∗ ) − 1 ( A − B ) .

Proof

This follows from arguments similar to those corresponding ones in [4, p. 229] and [7, p. 124].□

3 Determinantal inequalities for quaternion matrices

We are now in a position to state and prove our main results which extend inequalities (1.1), (1.4), and (1.5) to the corresponding ones for quaternion matrices.

Theorem 3.1

Let A , B ∈ H n × n . If I n − A ∗ A and I n − B ∗ B are both positive definite, then

(3.1) ∏ j = 1 k σ n − j + 1 2 ( I n − A ∗ B ) ≥ ∏ j = 1 k [ 1 − λ j ( A ∗ A ) ] [ 1 − λ j ( B ∗ B ) ] + ∏ j = 1 k [ 1 − λ j ( A ∗ A ) ] [ 1 − λ n − j + 1 ( A ∗ A ) ] − 1 σ n − j + 1 2 ( A − B )

for 1 ≤ k ≤ n .

Proof

Since ( I n − A ∗ B ) ∗ ( I n − A ∗ B ) is positive semidefinite, by Lemma 2.1, there exists a generalized unitary matrix U = ( u 1 , u 2 , … , u n ) ∈ H n × n such that

(3.2) U ∗ ( I n − A ∗ B ) ∗ ( I n − A ∗ B ) U = diag ( σ 1 2 ( I n − A ∗ B ) , σ 2 2 ( I n − A ∗ B ) , … , σ n 2 ( I n − A ∗ B ) ) .

Putting U k = ( u n − k + 1 , … , u n ) ∈ H n × k , by identity (2.6) in Lemma 2.6, we obtain

(3.3) U k ∗ ( I n − B ∗ A ) ( I n − A ∗ A ) − 1 ( I n − A ∗ B ) U k = U k ∗ ( I n − B ∗ B ) U k + U k ∗ ( A − B ) ∗ ( I n − A A ∗ ) − 1 ( A − B ) U k .

Taking determinants on both sides of (3.3) and using Lemma 2.3, we acquire

det ( U k ∗ ( I n − B ∗ A ) ( I n − A ∗ A ) − 1 ( I n − A ∗ B ) U k ) = det ( U k ∗ ( I n − B ∗ B ) U k + U k ∗ ( A − B ) ∗ ( I n − A A ∗ ) − 1 ( A − B ) U k ) ≥ det ( U k ∗ ( I n − B ∗ B ) U k ) + det ( U k ∗ ( A − B ) ∗ ( I n − A A ∗ ) − 1 ( A − B ) U k ) .

Applying inequalities (2.2) and (2.4) yields

det ( U k ∗ ( I n − B ∗ B ) U k ) ≥ ∏ j = 1 k λ n − j + 1 ( I n − B ∗ B ) = ∏ j = 1 k [ 1 − λ j ( B ∗ B ) ]

and

det ( U k ∗ ( A − B ) ∗ ( I n − A A ∗ ) − 1 ( A − B ) U k ) ≥ ∏ j = 1 k σ n − j + 1 2 ( ( A − B ) ∗ ( I n − A A ∗ ) − 1 ( A − B ) ) ≥ ∏ j = 1 k σ n − j + 1 2 ( A − B ) λ n − j + 1 ( ( I n − A ∗ A ) − 1 ) = ∏ j = 1 k σ n − j + 1 2 ( A − B ) [ 1 − λ n − j + 1 ( A ∗ A ) ] − 1 .

Using Lemma 2.4, inequality (2.3), and the matrix decomposition (3.2), we arrive at

det ( U k ∗ ( I n − B ∗ A ) ( I n − A ∗ A ) − 1 ( I n − A ∗ B ) U k ) = ∏ j = 1 k λ j ( U k ∗ ( I n − B ∗ A ) ( I n − A ∗ A ) − 1 ( I n − A ∗ B ) U k ) ≤ ∏ j = 1 k σ j 2 ( ( I n − A ∗ B ) U k ) λ j ( ( I n − A ∗ A ) − 1 ) = ∏ j = 1 k σ n − j + 1 2 ( I n − A ∗ B ) [ 1 − λ j ( A ∗ A ) ] − 1 .

The proof of Theorem 3.1 is complete.□

Corollary 3.1

Let A , B ∈ H n × n . If I n − A ∗ A and I n − B ∗ B are both positive definite, then

∏ j = 1 k σ n − j + 1 2 ( I n − A ∗ B ) ≥ ∏ j = 1 k [ 1 − λ j ( A ∗ A ) ] [ 1 − λ j ( B ∗ B ) ]

for 1 ≤ k ≤ n .

Proof

This follows from inequality (3.1) in Theorem 3.1.□

Corollary 3.2

Let A , B ∈ H n × n . If I n − A ∗ A and I n − B ∗ B are both positive definite, then

det ( I n − A ∗ A ) det ( I n − B ∗ B ) + ∣ det ( A − B ) ∣ 2 ≤ ∣ det ( I n − A ∗ B ) ∣ 2 .

In particular,

det ( I n − A ∗ A ) det ( I n − B ∗ B ) ≤ ∣ det ( I n − A ∗ B ) ∣ 2 .

Proof

This follows from letting k = n in Theorem 3.1.□

Theorem 3.2

Let A , B ∈ H n × n . For 1 ≤ k ≤ n , we have

∏ j = 1 k σ n − j + 1 2 ( I n − A B ∗ ) + ∏ j = 1 k σ n − j + 1 2 ( A + B ) ≤ ∏ j = 1 k [ 1 + λ n − j + 1 ( A ∗ A ) ] [ 1 + λ j ( B ∗ B ) ] .

Proof

There exists a generalized unitary matrix U = ( u 1 , u 2 , … , u n ) ∈ H n × n such that

U ∗ ( I n + A A ∗ ) U = diag ( λ 1 ( I n + A A ∗ ) , λ 2 ( I n + A A ∗ ) , … , λ n ( I n + A A ∗ ) ) .

Taking U k = ( u n − k + 1 , … , u n ) ∈ H n × k , utilizing the matrix identity (2.5) in Lemma 2.6, and employing inequalities in Lemmas 2.3 and 2.4 lead to

∏ j = 1 k [ 1 + λ n − j + 1 ( A ∗ A ) ] = det ( U k ∗ ( I n + A A ∗ ) U k ) ≥ det ( U k ∗ ( A + B ) ( I n + B ∗ B ) − 1 ( A + B ) ∗ U k ) + det ( U k ∗ ( I n − A B ∗ ) ( I n + B B ∗ ) − 1 ( I n − A B ∗ ) ∗ U k ) ≥ ∏ j = 1 k σ n − j + 1 2 ( A + B ) [ 1 + λ j ( B ∗ B ) ] − 1 + ∏ j = 1 k σ n − j + 1 2 ( I n − A B ∗ ) [ 1 + λ j ( B ∗ B ) ] − 1 .

The proof of Theorem 3.2 is thus complete.□

Corollary 3.3

Let A , B ∈ H n × n . Then

∣ det ( I n − A ∗ B ) ∣ 2 ≤ det ( I n + A ∗ A ) det ( I n + B ∗ B ) − ∣ det ( A + B ) ∣ 2 .

Proof

This follows from setting k = n in Theorem 3.2.□

qifeng618@hotmail.com qifeng618@qq.com https://qifeng618.wordpress.com

# Dedicated to Professor Jen-Chih Yao at China Medical University in Taiwan.

Acknowledgements

The authors appreciate anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

Funding information: This work was supported in part by the Natural Science Foundation of Inner Mongolia (Grant No. 2019MS01007), by the Science Research Fund of Inner Mongolia University for Nationalities (Grant No. NMDBY15019), and by the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (Grant No. NJZY19157 and No. NJZY20119) in China.
Author contributions: All authors contributed equally to the manuscript and read and approved the final manuscript.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Received: 2021-02-09

Revised: 2021-05-17

Accepted: 2021-06-21

Published Online: 2021-07-09

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Keywords for this article

determinantal inequality; matrix of quaternions; eigenvalue; singular value; positive definite matrix; Hua-Marcus-Zhang type

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