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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2021; 61(3): 583-590

Published online September 30, 2021

Copyright © Kyungpook Mathematical Journal.

### New Bounds for the Numerical Radius of a Matrix in Terms of Its Entries

Abdelkader Frakis

Department of Mathematics, Mustapha Stambouli University, Mascara, Algeria
e-mail : aekfrakis@yahoo.fr

Received: March 18, 2020; Revised: January 15, 2021; Accepted: February 8, 2021

In this work we give new upper and lower bounds for the numerical radius of a complex square matrix A using the entries and the trace of A.

Keywords: Numerical range, trace of matrix, Frobenius norm, numerical radius, spectral radius.

### 1. Introduction

The numericalrange of a complex $n×n$ matrix A is the set defined as

$W(A)={⟨Ax,x⟩, x∈ℂn, ‖x‖=1},$

where $⟨x,y⟩$ is the usual inner product of elements x and y in $ℂn$. The numerical range of the matrix A localizes its spectrum i.e $Λ(A)⊆W(A)$, where $Λ(A)$ denotes the spectrum of A. The numerical range has several properties.

The numerical radius $ω(A)$ is defined by

Numerous contributions related to numerical radius were made by various people including M. Goldberg, E. Tadmor and G. Zwas [1], also J. Merikoski and R. Kumar [4]. We cite here some properties of the numerical radius which are well known see [2]. Let A, B be two complex matrices and $α∈ℂ$,

1. $ω(A+B)≤ω(A)+ω(B),$

2. $ω(αA)=|α|ω(A),$

3. $ω(A)=ω(A∗),$

where $A∗$ is the conjugate transpose of A.

If M is any principle submatrix of A, then

$ω(M)≤ω(A).$

In this paper, without knowing thenumerical radius of the matrix A, we can estimate it by giving some upper and lower bounds using the entries and the trace of A.

Let A be a complex $n×n$ matrix with eigenvalues $λ1,⋯,λn$, the spectral radius of A is defined by

$ρ(A)=max1≤i≤n|λi|.$

It is well known, see [1], that

$ρ(A)≤ω(A)≤‖A‖≤2ω(A),$

where $‖A‖=max‖x‖=1‖Ax‖$ is the spectral norm.

Let $tr(A)=∑ i=1nλi$ denote the trace of A and let $su(A)=∑ i,j=1naij$ denote the sum of A.

Let ei be the column vector whose i-th component is equal to 1 while all the remaining components are 0.

Let R(A) and c denote the radius and center of the smallest disc $D$ which contains all eigenvalues of A.

In [3] C. R. Johnson gave an upper bound for the numerical radius

$ω(A)≤maxi∑ j=1n|aij|+|aji|2.$

J. K. Merikoski and R. Kumar [4] gave some lower bounds for the numerical radius $ω(A)$ for example :

$maxi|aii|≤ω(A)$

and

$su(A)n≤ω(A).$

### 2. Bounds For the Numerical Radius

In this section, we give some upper and lowers bounds for the numerical radius of a given complex $n×n$ matrix.

Proposition 2.1. For any matrix A, we have

$R(A)≤ω(A).$

Theorem 2.2. Let A=(aij) be a normal $n×n$ matrix, we have

$maxi≠j|aij|≤ω(A).$

Proof. Let z be any complex number. For i ≠ j,

$|aij|=|ei∗(A−zI)ej|≤‖ei‖.‖(A−zI)ej‖=‖(A−zI)ej‖ ≤sup‖u‖=1‖(A−zI)u‖=maxi|λi−z|.$

Since $infzmaxi|λi−z|=R(A)$, then $maxi≠j|aij|≤ω(A).$

Corollary 2.3. Let A=(aij) be a normal

$n×n$ matrix, we have

$12maxi≠j(|aij|+|aji|)≤ω(A).$

Proof. Applying the result of the above theorem to the matrix $zA+z¯A∗2$, where $z∈ℂ$ with |z|=1, it follows that $12maxi≠j|zaij+z¯aji¯|≤ω(A)$. Since $max|z|=1|zaij+z¯aji¯|=|aij|+|aji|$, then the required result is obtained.

Theorem 2.4. Let A=(aij) be a complex $n×n$ matrix, we have

$ω(A)≤maxi|aii|+(n−1)maxi≠j|aij|.$

Proof. Write $x=(x1,x2,⋯,xn)$ and let $λ∈W(A)$ then $λ=xAx∗$ with $‖x‖=1$. Hence $λ=∑ i,jaijxjxi∗$, thus $|λ|≤∑ i,j|aij|ξiξj$ where $ξi=|xi|$. It follows that

$|λ|≤∑i|aii|ξi2+∑ i≠j|aij|ξjξi ≤maxi|aii|+maxi≠j|aij|∑ i

We have used the fact that $2ξiξj≤ξi2+ξj2$. Since $ω(A)=maxλ∈W(A)|λ|$, then this completes the proof.

Corollary 2.5. Let A=(aij) be a complex $n×n$ matrix, we have

$ω(A)≤nmaxi,j|aij|.$

Theorem 2.6. Let A=(aij) be a complex $n×n$ matrix, we have

$ω(A)≤maxi|aii|+∑ i≠j|aij |21/2.$

Proof. Let $λ∈W(A)$ then $λ=∑ i,jaijxjxi∗$. Hence $|λ|≤∑ i,j|aij|ξiξj$ where $ξi=|xi|$. It follows that $|λ|≤∑i|aii|ξi2+∑ i≠j|aij|ξjξi$. Rewriting $|aij|ξjξi$ as $|aij|×ξjξi$ and applying the Cauchy Schwarz's inequality, we obtain

$|λ|≤maxi|aii|+ ∑i≠j| a ij |21/2 ∑i≠j ξ i2ξj21/2 ≤maxi|aii|+ ∑i≠j| a ij |21/2 ∑i ξ i2.∑ j ξj21/2 ≤maxi|aii|+ ∑i≠j| a ij |21/2.$

Since $ω(A)=maxλ∈W(A)|λ|$, then the desired result is obtained.

Let A=(aij) be a complex $n×n$ matrix and let $Li=∑j|aij|−|aii|, Cj=∑ i|aij|−|ajj|$.

Theorem 2.7. Let A=(aij), Li and Cj be as described above and let $L=max(Li), C=max(Cj)$. Then

$ω(A)≤maxi|aii|+(LC)1/2.$

Proof. Let $λ∈W(A)$ then $λ=∑ i,jaijxjxi∗$. Hence $|λ|≤∑ i,j|aij|ξiξj$ where $ξi=|xi|$. Thus $|λ|≤∑i|aii|ξi2+∑ i≠j|aij|ξjξi$.

Rewriting $|aij|ξjξi$ as $|aij|1/2ξi×|aij|1/2ξj$ and applying the Cauchy Schwarz's inequality, it follows that

$|λ|≤maxi|aii|+∑i≠j|aij|ξi21/2∑i≠j|aij|ξj21/2 =maxi|aii|+∑iLiξi21/2∑jCjξj21/2 ≤maxi|aii|+L∑iξi21/2C∑jξj21/2 =maxi|aii|+(LC)1/2.$

Since $ω(A)=maxλ∈W(A)|λ|$, then the assertion follows immediately.

Theorem 2.8. Let $A=(aij)$, Li and Ci be as described above and let $Si=Li+Ci2, S=maxiSi$. Then

$ω(A)≤maxi|aii|+S.$

Proof. Let $λ∈W(A)$ then $λ=∑ i,jaijxjxi∗$. Hence $|λ|≤∑ i,j|aij|ξiξj$ where $ξi=|xi|$. It follows that

$|λ|≤∑i|aii|ξi2+∑ i≠j|aij|ξjξi ≤maxi|aii|+12∑ i≠j|aij|(ξi2+ξj2) =maxi|aii|+12∑iL iξi2+12∑ jCjξj2 =maxi|aii|+∑iS iξi2 ≤maxi|aii|+S.$

Since $ω(A)=maxλ∈W(A)|λ|$, then the result follows directly.

Lemma 2.9. If $z1,⋯,zn$ are complex numbers, then

$z1+⋯+znn≤maxi|zi|.$

Corollary 2.10. Let A be a complex $n×n$ matrix with eigenvalues $λ1,⋯,λn$. Then

$tr(A)n≤ω(A).$

Proof. Using the previous lemma, $zi=λi$, it follows that $tr(A)n≤ρ(A)≤ω(A).$

Theorem 2.11. Let A=(aij) be a complex

$n×n$ matrix. Then

$maxi≠jaii+ajj−aij−aji2≤ω(A).$

Proof. For i ≠ j, we have $(ei−ej)∗A(ei−ej)2=aii+ajj−aij−aji2≤ω(A)$.

Theorem 2.12. Let A=(aij) be a complex $n×n$ matrix. Then

$nn−1tr(A)n−su(A)n2≤ω(A).$

Proof. Using Lemma 2.9. where the z's are the n(n-1) numbers $zij=aii+ajj−aij−aji2,i≠j,$ thus

$maxi≠jaii+ajj−aij−aji2≥1n(n−1)∑i∑ j≠iaii+ajj−aij−aji2 =1n(n−1)n∑ i=1naii−∑ i,j=1naij =nn−1tr(A)n−su(A) n2.$

Using the previous theorem then the required statement follows immediately.

Theorem 2.13. Let Abe a complex $n×n$ matrix with eigenvalues $λ1,⋯,λn$. Then

$1n∑ i=1 n|λi|2−|tr(A)|2n12≤ω(A).$

Proof. We have

$∑ i=1n|λi−c|2≤nR2(A),$

where c and R(A) are the center and the radius of the smallest disc $D$, respectively. On the other hand,

$∑i=1n|λi−c|2=∑i=1n |λi|2−c λi ¯−c¯λi+|c|2 =∑i=1n|λi|2−|tr(A)|2n+nc−tr(A)n 2.$

It is clear that the choice $c=tr(A)/n$ gives the smallest possible value for this last expression. Hence $1n∑ i=1 n|λi|2−|tr(A)|2n≤R2(A)≤ω2(A).$

Corollary 2.14. Let A be a normal $n×n$ matrix. Then

$1n‖A‖Fr2−|tr(A)|2n12≤ω(A),$

where $‖A‖Fr2=∑ i,j=1n|aij|2=trAA∗$ is the Frobenius norm.

Proof. Since A is normal, then $∑ i=1n|λi|2=‖A‖Fr2$. Hence the desired result follows.

Theorem 2.15. Let A=(aij) be a Hermitian

$n×n$ matrix. Then

$12maxi≠jaii+ajj+(aii−ajj)2+4|aij|2≤ω(A).$

Proof. Let M be any principal submatrix of A. Let $1≤i and

$M=aiiaijajiajj,$

then

$ρ(M)=12aii+ajj+(aii−ajj)2+4|aij|2≤ω(M)≤ω(A).$

### 3. The Areal Numerical Radius of Matrices

Let $Γ(A)$ denotes the area of the smallest disc $D$ which contains all eigenvalues of the matrix A.

R. A. Smith and L. Mirsky in [5] called areal spread of the matrix A the ratio $σ(A)‖A‖2$ where $σ(A)$ is the minimal area in the complex plane and $‖.‖$ is the euclidean matrix norm. In analogy with this concept, let $Γ(A)ω2(A)$ be the areal numerical radius of A. In the following theorem we give an estimate to the supremum of the areal numerical radius of A as A ranges over all nonzero $n×n$ matrices.

Theorem 3.1. Let A be a complex $n×n$ matrix and let $Γ(A)$ be as described above. Then

$supΓ(A)ω2(A)=π,$

where the supremum is taken over all nonzero $n×n$ matrices A.

Proof. Since $Γ(A)=πR2(A)$, it is sufficient to prove that $supR(A)ω(A)=1$. We have $R(A)≤ρ(A)≤ω(A)$, on the other hand, taking $A=diag(−1,0,⋯,0,1),$ it follows that R(A)=1 and ω(A)=1. Hence $supR(A)ω(A)=1$, this completes the proof.

### Acknowledgements.

The author would like to thank the reviewers for their very helpful comments and suggestions.

1. M. Goldberg, E. Tadmor and G. Zwas, The numerical radius and spectral matrices, Linear and Multilinear Algebra, 2(1975), 317-326.
2. R. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
3. C. R. Johnson, A gersgorin inclusion set for the field of values of a finite matrix, Proceedings of the american mathematical society. 41(1973), 57-60.
4. J. K. Merikoski and R. Kumar, Lower bounds for the numerical radius, Linear Algebra Appl., 410(2005), 135-142.
5. R. A. Smith and L. Mirsky, The areal spread of matrices, Linear Algebra Appl., 2(1969), 127-129.