EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 1, 2024, 243-247 ISSN 1307-5543 – ejpam.com Published by New York Business Global Matrix mixed inequalities Feng Zhang1,∗, Chunwen Zhang2 1 Department of Mathematics, Southeast University Chengxian College, Nanjing 210000, China 2 Department of Electrical and Computer Engineering, Southeast University Chengxian College, Nanjing 210000, China Abstract. In this paper, we prove that all the eigenvalues of arbitrarily complex matrix are located in one closed disk, which is a refinement of some existing inequalities. 2020 Mathematics Subject Classifications: 47A30, 15A42, 15A18 Key Words and Phrases: Disk, Majorization, unitarily invariant norms, eigenvalue; location 1. Introduction We denote by Mn the vector space of all complex n× n matrices. The notation A ≥ 0 is used to mean that A is positive semidefinite. For A ∈ Mn, the conjugate transpose of A is denoted by A∗. Denote by λj (A) (1 ≤ j ≤ n) the class of all eigenvalues of A ∈ Mn and ∥A∥F = √ tr (AA∗), [A,B] = AB − BA. The singular values of A are enumerated as s1 (A) ≥ s2 (A) ≥ · · · ≥ sn (A). These are the eigenvalues of the positive semidefinite matrix |A| = (A∗A) 1 2 . The estimation and location of eigenvalues are always hot topics of matrix analysis [1], [2]. It plays an important role in many fields of applied science. Let M ∈ Mn be an n×n complex matrix partitioned as M = [ Ak Bk,n−k Cn−k,k Dn−k ] , where 1 ≤ k ≤ n− 1. The following estimation n∑ i=1 |λi|2 ≤ ∥M∥2F − max 1≤k≤n−1 ( ∥Bk,n−k∥F − ∥Cn−k,k∥F )2 is an elegant result on eigenvalues due to Tu [3]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.5009 Email addresses: fzhang1024@163.com (F. Zhang), 2153228327@qq.com (C. Zhang) https://www.ejpam.com 243 © 2024 EJPAM All rights reserved. F. Zhang, C. Zhang / Eur. J. Pure Appl. Math, 17 (1) (2024), 243-247 244 In [4], Gu proposed a new idea which uses only one single closed disk to locate eigen- values of a given n×n complex matrix. He proved that all the eigenvalues of any complex matrix A are located in the following disk:∣∣∣∣λj − trA n ∣∣∣∣ ≤ ( n− 1 n ( ∥A∥2F − |trA|2 n )) 1 2 (1) for j = 1, 2, · · · , n. Zou et al. [5] showed that all eigenvalues of M are located in the following disk:{ z ∈ C : ∣∣∣∣z − trM n ∣∣∣∣ ≤ √ ∥M∥2F − |trM |2 n − max 1≤k≤n−1 ( ∥Bk,n−k∥F − ∥Cn−k,k∥F )2} . (2) Let M (x) = [ Ak xBk,n−k x−1Cn−k,k Dn−k ] , where Ak is a k × k principal submatrix of M (1 ≤ k ≤ n− 1) and x is any non-zero real number. For convenience, we write, respectively. △M (k, x) = ∥M∥2F − [( 1− x2 ) ∥Bk,n−k∥2F + ( 1− x−2 ) ∥Cn−k,k∥2F ] − |trM |2 n and fM (k, x) = ( (△M (k, x))2 − 1 2 ∥[M (x) ,M (x)∗]∥2F ) 1 2 + |trM |2 n . In [6], Wu et al. proved that∣∣∣∣λj(M)− trM n ∣∣∣∣ ≤ min x̸=0 min 1≤k≤n−1 √ n− 1 n ( fM (k, x)− |trM |2 n ) 1 2 , (3) which is a refinement of inequality (2). It is natural to ask whether stronger inequality of (2) might be proved. This is a part of the motivation for our study. 2. Main result We let the symbol Sl denote the set {1, · · ·n}\{l} for l = 1, 2, · · · , n. In this section, a sharper estimation of the eigenvalues is presented. In order to obtain our result, we need the following lemmas. Lemma 1. [7] Let A ∈ Mn with n ≥ 3, then∣∣∣∣λl(A)− trA n ∣∣∣∣2 ≤ n− 1 n  n∑ j=1 |λj(A)|2 − |trA|2 n − 1 2 s2(A)  for l = 1, 2, · · · , n and s(A) = min 1≤l≤n max j,k∈Sl |λj(A)− λk(A)|. F. Zhang, C. Zhang / Eur. J. Pure Appl. Math, 17 (1) (2024), 243-247 245 Lemma 2. [7] Let A ∈ Mn, then n∑ j=1 |λj(A)|2 ≤ √√√√(∥A∥2F − |trA|2 n )2 − ∥[A,A∗]∥2F 2 + |trA|2 n . Next we give a new proof of Lemma 2.2 in [6], which plays a key role in their discussion. Lemma 3. Let M = [ Ak Bk,n−k Cn−k,k Dn−k ] with eigenvalues λ1, λ2, · · · , λn, then n∑ j=1 |λj |2 ≤ min x ̸=0 min 1≤k≤n−1 fM (k, x) is valid for any non-zero number x. Proof. Let X = [ xIk 0 0 In−k ] , then M(x) = XMX−1, where Ik is a k × k unit matrix. Obviously, M(x) is similar to M . By Lemma [2], we have n∑ j=1 |λj(M)|2 = n∑ j=1 |λj(M(x))|2 (4) ≤ √√√√(∥M(x)∥2F − |trM(x)|2 n )2 − ∥[M(x),M(x)∗]∥2F 2 + |trM(x)|2 n = √√√√(∥M(x)∥2F − |trM |2 n )2 − ∥[M(x),M(x)∗]∥2F 2 + |trM |2 n , where ∥M(x)∥F = ( ∥M∥2F − [( 1− x2 ) ∥Bk,n−k∥2F + ( 1− x−2 ) ∥Cn−k,k∥2F ]) 1 2 . (5) Combing inequality (4) and equality (5), we conclude Lemma 3. We now focus on the location of the eigenvalues of complex matrices. Theorem 1. Let M = [ Ak Bk,n−k Cn−k,k Dn−k ] with eigenvalues λ1, λ2, · · · , λn ( n ≥ 3), then all of eigenvalues of M are included by the following disk: ∣∣∣∣λl(M)− trM n ∣∣∣∣ ≤ min x ̸=0 min 1≤k≤n−1 √ n− 1 n ( fM (k, x)− |trM |2 n − 1 2 s2(M) ) 1 2 for l = 1, 2, · · · , n and s(M) = min 1≤l≤n max j,k∈Sl |λj(M)− λk(M)|. REFERENCES 246 Proof. Combining Lemmas 2.1 and 2.3, we deduce that∣∣∣∣λl(M)− trM n ∣∣∣∣2 ≤ n− 1 n  n∑ j=1 |λj(M)|2 − |trM |2 n − 1 2 s2(M)  ≤ n− 1 n ( min x ̸=0 min 1≤k≤n−1 fM (k, x)− |trM |2 n − 1 2 s2(M) ) ≤ min x̸=0 min 1≤k≤n−1 n− 1 n ( fM (k, x)− |trM |2 n − 1 2 s2(M) ) . Therefore,∣∣∣∣λl(M)− trM n ∣∣∣∣ ≤ min x ̸=0 min 1≤k≤n−1 √ n− 1 n ( fM (k, x)− |trM |2 n − 1 2 s2(M) ) 1 2 for s(M) = min 1≤l≤n max j,k∈Sl |λj(M)− λk(M)|. This completed the proof. For complex matrix with order n(n > 2), then the computation of Theorem 2.4 requires approximately n3 2 additional calculations compared to the computation of inequality(3). This indicates that its computational complexity is greater than the computational com- plexity in (3). But, in theory, Theorem 2.4 is a refinement of (3). 3. Funding Feng Zhang is supported by the Southeast University Chengxian College Young Teacher Research Development Fund Projection (No. z0058). Acknowledgements Feng Zhang thanks Shiyu Xu for her kindly help and love her forever. References References [1] R. A. Horn and C. R. Johnson, Matrix Analysis. 1985. [2] R. Bhatia, Positive Definite Matrices.Princeton University Press, 2007. [3] B. Tu. The lower bound of the rank and non-singularity of matrices. Coll. J. Fu Dan Univ., 416–422, 1982. [4] Y. Gu. The distribution of eigenvalues of a matrix, Acta Math. Appl. Sinica, 501–511, 1994. REFERENCES 247 [5] L. Zou, Y. Jiang. Estimation of the eigenvalues and the smallest singular value of matrices, Linear Algebra Appl., 1203–1211, 2010. [6] J. Wu, J. Zhao, A survey of the progress of locating methods of complex matrices eigenvalues and some new location theorem and their applications, IMA J. Appl. Math., 273–285, 2015. [7] I. H. Gumus, O. Hirzallah, and F. Kittaneh, Eigenvalue localization for complex matrices, Electronic Journal of Linear Algebra ,892–906, 2014.