Let M_n be the space of complex matrices. For A ∈ M_n, A^∗ is the conjugate transpose matrix of A, λ₁, λ₂, …, λ_n are the eigenvalues of A repeated according to their algebraic multiplicities and is the Frobenius norm. and .

In 1931, Geršchgorin proved the eigenvalues of A ∈ M_n are included by the union of n small disks [1]:

where

Next, Ostrowski has made improvements on this [2]. Brauer used Cassini ovals to determine inclusion regions of any n × n complex matrix’s eigenvalues [1]. Fan [3] obtained an inequality that is similar to Geršchgorin’s.

In [4–15], the authors used the union of some subsets (small disks, ovals, or other domains) to determine the location of eigenvalues.

Specially, motivated by Rojo’s work in [9], in [15], Wu and Zhao proved that

Theorem 1. Let A ∈ M_n with eigenvalue λ. Then,

()

and

()

where

An application of Theorem 1 is as follows:

Theorem 2. Let A ∈ M_n with eigenvalues λ₁, λ₂, …, λ_n. Then, all its eigenvalues are lie in the elliptic region

()

where

As Wu and Zhao stated in [15], such an elliptic region is the best one.

Matrix eigenvalues are widely used in quantum mechanics (see [16–21] for more details). Therefore, studying the estimation of matrix eigenvalues is meaningful in physics. For the eigenvalues of a 2 × 2 matrix, their range could be considered by direct calculations. Therefore, mathematicians always consider the order of the matrix is greater than 2 for better applications. After reading [15], our concern is whether stronger versions of inequality (Equation (2)) and equality (Equation (3)) could be found for n ≥ 3. This is the motivation for studying this paper. In this paper, we give some refinements of inequalities (Equations (1) and (2)) and equality (Equation (3)).

2. Main Results

We start our work with the following auxiliary lemmas. Throughout the paper, we write S_ℓ to present the set {1, 2, …, n}∖{ℓ} for ℓ = 1, 2, …, n..

Lemma 1 see [14]. Let z_i(1 ≤ i ≤ n) be complex numbers with n ≥ 3. Then,

()

for ℓ = 1, 2, …, n.

Lemma 2 see [22]. Let A ∈ M_n with eigenvalue λ_i(1 ≤ i ≤ n). Then,

()

where

Lemma 3 see [14]. Let z₁, z₂, …, z_n be complex numbers. Then,

()

From the above lemmas, we conclude our first result.

Theorem 3. Let A ∈ M_n(n ≥ 3) with eigenvalue λ_ℓ. Then,

and

where

and

Proof. Let λ_ℓ be an arbitrary eigenvalue of A. By equality (Equation (4)), we have the following:

where the first inequality is due to Equation (7) and the second inequality follows from the inequality (Equation (5)) and the definition of s²(A).

With the same method, we get the following:

Corollary 1. Let A ∈ M_n(n ≥ 3) with eigenvalue λ_ℓ. Then,

and

where

The next result is a consequence of Theorem 3.

Theorem 4. Let A ∈ M_n with eigenvalues λ₁, λ₂, …, λ_n(n ≥ 3). Then, all eigenvalues are located in elliptic region

for some l with 0 < l ≤ 1, where

Proof. We first consider the case tr(A) = 0. It follows from Theorem 3 that

and

The real parts and imaginary parts of eigenvalue λ_ℓ are satisfying the following:

We rephrased it as follows:

For the general case, we set

and repeated the above progress. Therefore,

Therefore, the desired result is obtained.

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

This work is supported by the National Research Project Incubation Foundation of Southeast University Chengxian College (2022NCF008) and the Young Teachers’ Research and Development Fund of Southeast University Chengxian College (z0058).

Open Research

Data Availability Statement

Data sharing is not applicable to this manuscript as no new data were created or analyzed in this study.

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Eigenvalue Localization Inequalities for Complex Matrices With Order n ≥ 3

Abstract

1. Introduction

2. Main Results

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

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Eigenvalue Localization Inequalities for Complex Matrices With Order n ≥ 3

Abstract

1. Introduction

2. Main Results

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

References

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