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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.

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

W(A)={Ax,x,xn,x=1},

where x,y is the usual inner product of elements x and y in n. 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

ω(A)=supλW(A)|λ|   or ω(A)=maxx=1|Ax,x|.

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,

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)=max1in|λi|.

It is well known, see [1], that

ρ(A)ω(A)A2ω(A),

where A=maxx=1Ax 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)nω(A).

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

maxij|aij|ω(A).

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

|aij|=|ei(AzI)ej|ei.(AzI)ej=(AzI)ej        supu=1(AzI)u=maxi|λiz|.

Since infzmaxi|λiz|=R(A), then maxij|aij|ω(A).

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

n×n matrix, we have

12maxij(|aij|+|aji|)ω(A).

Proof. Applying the result of the above theorem to the matrix zA+z¯A2, where z with |z|=1, it follows that 12maxij|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|+(n1)maxij|aij|.

Proof. Write x=(x1,x2,,xn) and let λW(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+ ij|aij|ξjξimaxi|aii|+maxij|aij| i<j2ξ iξjmaxi|aii|+maxij|aij|(n1)iξ i2=maxi|aii|+(n1)maxij|aij|.

We have used the fact that 2ξiξjξi2+ξj2. Since ω(A)=maxλW(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|+ ij|aij |21/2.

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

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

Since ω(A)=maxλW(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 λW(A) then λ= i,jaijxjxi. Hence |λ| i,j|aij|ξiξj where ξi=|xi|. Thus |λ|i|aii|ξi2+ ij|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|+ij|aij|ξi21/2ij|aij|ξj21/2=maxi|aii|+iLiξi21/2jCjξj21/2maxi|aii|+Liξi21/2Cjξj21/2=maxi|aii|+(LC)1/2.

Since ω(A)=maxλW(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 λW(A) then λ= i,jaijxjxi. Hence |λ| i,j|aij|ξiξj where ξi=|xi|. It follows that

|λ|i|aii|ξi2+ ij|aij|ξjξimaxi|aii|+12 ij|aij|(ξi2+ξj2)=maxi|aii|+12iL iξi2+12 jCjξj2=maxi|aii|+iS iξi2maxi|aii|+S.

Since ω(A)=maxλW(A)|λ|, then the result follows directly.

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

z1++znnmaxi|zi|.

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

tr(A)nω(A).

Proof. Using the previous lemma, zi=λi, it follows that tr(A)nρ(A)ω(A).

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

n×n matrix. Then

maxijaii+ajjaijaji2ω(A).

Proof. For i ≠ j, we have (eiej)A(eiej)2=aii+ajjaijaji2ω(A).

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

nn1tr(A)nsu(A)n2ω(A).

Proof. Using Lemma 2.9. where the z's are the n(n-1) numbers zij=aii+ajjaijaji2,ij, thus

maxijaii+ajjaijaji21n(n1)i jiaii+ajjaijaji2          =1n(n1)n i=1naii i,j=1naij          =nn1tr(A)nsu(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)|2n12ω(A).

Proof. We have

i=1n|λic|2nR2(A),

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

i=1n|λic|2=i=1n |λi|2c λi ¯c¯λi+|c|2    =i=1n|λi|2|tr(A)|2n+nctr(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)|2nR2(A)ω2(A).

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

1nAFr2|tr(A)|2n12ω(A),

where AFr2= i,j=1n|aij|2=trAA is the Frobenius norm.

Proof. Since A is normal, then i=1n|λi|2=AFr2. Hence the desired result follows.

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

n×n matrix. Then

12maxijaii+ajj+(aiiajj)2+4|aij|2ω(A).

Proof. Let M be any principal submatrix of A. Let 1i<jn and

M=aiiaijajiajj,

then

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

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)A2 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.

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

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