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

Published online September 30, 2021

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

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

$W(A)={⟨Ax,x⟩, x∈ℂn, ‖x‖=1},$

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 , also J. Merikoski and R. Kumar . We cite here some properties of the numerical radius which are well known see . 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 , 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  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  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  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.

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

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