Localization theorems for eigenvalues of quaternionic matrices

arXiv:1502.08014v1 [math.RA] 17 Jan 2015

Localization theorems for eigenvalues of quaternionic matrices Sk. Safique Ahmad∗

Istkhar Ali†

March 2, 2015

Abstract Ostrowski type and the corrected version of the Brauer type theorems are derived for the left eigenvalues of quaternionic matrix. Some distribution theorems are given in terms of ovals of Cassini that are sharper than the Ostrowski type theorems, respectively, for left and right eigenvalues of quaternionic matrix. In addition, generalizations of the Gerschgorin type theorems are discussed for both the left and right eigenvalues of quaternionic matrix, and finally, we show that our framework is so developed that generalizes the existing results in the literatures.

Keywords. Quaternionic matrices, left and right eigenvalue, Gerschgorin type theorems, Brauer type theorem. AMS subject classification. 15A18, 15A66.

1

Introduction

Localization theorems of quaternionic matrices have been becoming a wide range of research by several authors due to its various applications in the fields of Science and Engineering, for instance, see e.g., [1, 2, 6, 8, 9, 11, 14, 15, 17, 18, 20–22] and references therein. Gerschgorin type theorems for quaternion matrices were proposed by Zhang [22]. Ostrowski and Brauer type theorems have been found, see e.g., [12, 23]. The above localization theorems are well known ∗

Corresponding author: School of Basic Sciences, Discipline of Mathematics, IET DAVV Campus Khandwa

Road Indore, MP 452017; email: [email protected], Phone: +91-731-2438731, Fax: +91-731-2364182 † School of Basic Sciences, Discipline of Mathematics, Indian Institute of Technology, Indore, IET, DAVV Campus, Khandwa Road, Indore-452017, India, email: the CSIR, Govt. of India.

1

[email protected]. Research work funded by

in the literature for the case of complex field [3,7,10,16,19]. Unlike complex field, two different Gerschgorin type theorems have been found due to non-commutativity of quaternions [22]. Similarly, other localization theorems are of two kinds as two different left and right eigenvalues exist incase of quaternionic matrix. Ostrowski and Brauer type theorems have been found in [23] for the right eigenvalues of quaternionic matrix with real diagonal entries and the Brauer type theorem for the left eigenvalues of quaternionic matrix has been derived in [12]. But there is no literature of the Ostrowski type theorem for the left eigenvalues and further, the Brauer type theorem that has been derived in [12] for the left eigenvalues is found to be incorrect for the case of deleted absolute column sums of quaternionic matrix. In this paper we provide a general setup on localization theorems and to derive the Ostrowski type theorem for the left eigenvalues of quaternionic matrix. Derivation of localization theorems for the left and right eigenvalues are derived in terms of ovals of Cassini that provide better estimation than the Ostrowski type theorems for the left and right eigenvalues of a quaternionic matrix, respectively, which generalize some existing results on this direction. Consider a matrix A := (aij ) ∈ Mn (H), where Mn (H) is the set of n-by-n quaternionic matrices. We present the Ostrowski type theorem for left eigenvalues, i.e., all the left eigenvalues of A are located in the union of n balls Ti (A) := {z ∈ H : |z − aii | ≤ Riγ Ci1−γ }, P P where γ ∈ [0, 1], Ri := ni6=j=1 |aij |, and Ci := ni6=j=1 |aji |. As a consequence, we provide the sufficient conditions for a quaternionic matrix A to be nonsingular. In 2008, Junliang et

al. [12] have been presented Brauer type theorem for the left eigenvalues, see [12, Theorems 4, 5]. However, [12, Theorem 5] seems to be incorrect, follows from a counter example given in Section 3, and further, we derive the corrected version of it. In addition, we show the stronger results than [12, Theorems 6, 7] and a sharper result to the existing result given in [23, Theorem 4.3]. Moreover, we obtain the generalizations of some existing results given in [5,20,22] for the case of generalized H¨ older’s inequality, i.e., all the left eigenvalues of A ∈ Mn (H) are con1−γ

tained in the union of n generalized balls: Bi (A) := {z ∈ H : |z − aii | ≤ (n − 1) q Riγ Ni1−γ }, P 1 p n p , 1p + 1q = 1, p, q ∈ (1, ∞). In case of right where γ ∈ [0, 1] and Ni := i6=j=1 |aij |

eigenvalues, the above result is not true, while we show that for every right eigenvalue λ of

A ∈ Mn (H) there exists a non-zero quaternion β such that β −1 λβ is contained in the union  of n generalized balls Bi (A), i.e., ρ−1 λρ : 0 6= ρ ∈ H ∩ ∪ni=1 Bi (A) 6= ∅. As a consequence, we find that all the right eigenvalues of A ∈ Mn (H) with real diagonal entries are contained 2

in the union of n generalized balls Bi (A), and finally, we also present the upper bounds of absolute sum of the left and right eigenvalues for a quaternionic matrix A which generalizes the existing results given in [13].

2

Notation and preliminaries

Throughout the paper, we adopt the following notations. We denote R and C be the fields of real and complex numbers, respectively. The set of real quaternions is defined by H := {q = a0 + a1 i + a2 j + a3 k : a0 , a1 , a2 , a3 ∈ R} with i2 = j2 = k2 = ijk = −1. For q ∈ H, we p denote the conjugate of q by q := a0 − a1 i − a2 j − a3 k and |q| := a20 + a21 + a22 + a23 as the modulus of q. Let Hn be the right vector space over H. For x, y ∈ Hn , define hx, yi := y H x as an p inner product and kxk := hx, xi, the norm on Hn . Define Mn (R), Mn (C), and Mn (H) be the

set of n-by-n real, complex, and quaternionic matrices, respectively. For A ∈ Mn (K), K := {R, C, H}, the transpose and conjugate transpose are given by AT and AH , respectively. We

denote [q] the equivalence class containing q ∈ H. We write A = A1 + A2 j, for (H),  A ∈ Mn A1 A2  as where A1 , A2 ∈ Mn (C) and define Ψ : Mn (H) → M2n (C) by ΨA := Ψ(A) :=  −A2 A1 adjoint complex matrix. Definition 2.1 Let A ∈ Mn (H). Then the left, right, and the standard eigenvalues, respectively, are given by Λl (A) := {λ ∈ H : Ax = λx for some non-zero x ∈ Hn } , Λr (A) := {λ ∈ H : Ax = xλ for some non-zero x ∈ Hn } , and Λs (A) := {λ ∈ C : Ax = xλ for some non-zero x ∈ Hn , ℑ(λ) ≥ 0} . Lemma 2.2 [4]. Let A ∈ Mn (H) and let λ be a right eigenvalue of A, then ρ−1 λρ, is also a right eigenvalue of A, where 0 6= ρ ∈ H. Theorem 2.3 [21, Theorem 4.3]. Let A ∈ Mn (H). Then the following statements are equivalent: (a) A is invertible, (b) Ax = 0 has a unique solution, (c) det(ΨA ) 6= 0, (d) ΨA is invertible, (e) A has non-zero eigenvalues either left or right.

3

We now extend H¨ older’s inequality from complex field to quaternion field which is as follows, however generalized Cauchy Schwartz’s inequality on quaternion vectors has been proved in [20]. Lemma 2.4 (Generalized H¨ older’s inequality) For arbitrarily quaternion vectors z := (z1 , . . . , zn ) ∈ Hn , and w := (w1 , . . . , wn ) ∈ Hn , the following inequality holds: n X k=1

|zk wk | ≤

n X k=1

!1

n X

p

p

|zk |

k=1

!1 q

q

|wk |

,

1 1 + = 1, p, q ∈ (1, ∞). p q 1

1

Proof. Let α := (|z1 |p + |z2 |p + · · · + |zn |p ) p and β := (|w1 |q + |w2 |q + · · · + |wn |q ) q . Using Young’s inequality ab ≤

ap p

+

bq q ,

where a, b are nonzero positive numbers in R, and for

p, q ∈ (1, ∞), we write n X |zk wk | k=1

αβ

=

  n  X |zk | |wk | k=1

α

β

n

n

k=1

k1

1 1 1 X |zk |p 1 X |wk |q + ≤ + = 1. ≤ p αp q βq p q

Thus n X k=1

3

|zk wk | ≤

n X k=1

!1

p

|zk |p

n X k=1

!1 q

|wk |q

. 

Distribution for left and right eigenvalues of quaternionic matrix

Let A := (aij ) ∈ Mn (H) and define the deleted absolute row and column sums: Ri :=

n X

i6=j=1

|aij | and Ci :=

n X

i6=j=1

|aji |.

Throughout this section we would be using the above notations in our theory. It has been proved in [23] that A and AH have the same right eigenvalues. However, this is not true for the case of left eigenvalues which follows from the following example.     i 0 −i 0  . Then AH =  . Example 3.1 Consider A =  0 j 0 −j

This example shows, A and AH have different left eigenvalues. We now give the following theorem for left eigenvalues of A and AH .

4

Theorem 3.2 Let A ∈ Mn (H) and λ be a left eigenvalue of A. Then λ is the left eigenvalue of AH . Proof. Let λ be a left eigenvalue of A, then by definition, there exists 0 6= x ∈ Hn such that (A − λI)x = 0 if and only if Ψ(A−λI) Ψx = 0. It follows that λ is a left eigenvalue of A if   and only if det Ψ(A−λI) = 0. Then the following results hold: h i λ is a left eigenvalue of A if and only if det ΨH =0 h (A−λI) i λ is a left eigenvalue of A if and only if det Ψ(A−λI)H = 0 i h λ is a left eigenvalue of A if and only if det Ψ(AH −λI) = 0.

Thus λ is a left eigenvalue of AH . 

It has been found from the literature that the Gerschgorin type theorem for left eigenvalues of matrix A ∈ Mn (H) in terms of deleted absolute row sums [22]. However, there is no literature on the Gerschgorin type theorem for left eigenvalues in terms of deleted absolute column sums of A. To derive the generalized Ostrowski theorem, we need Gerschgorin type theorems for deleted absolute row and column sums of A. So the derivation of Gerschgorin type theorem is required for deleted absolute column sums of A which is as follows. Theorem 3.3 Let A := (aij ) ∈ Mn (H). Then all the left eigenvalues of A are located in the union of n Gerschgorin balls Ωi (A) := {z ∈ H : |z − aii | ≤ Ci } , i.e., Λl (A) ⊆ Ω(A) := ∪ni=1 Ωi (A). Proof. Let λ be a left eigenvalue of A. Then by Theorem 3.2, λ is a left eigenvalue of AH . Thus AH x = λx for some non-zero x := [x1 , . . . , xn ]T ∈ Hn . Let xt be an element of x such that |xt | ≥ |xi | ∀ i (1 ≤ i ≤ n). Then |xt | > 0. From t-th equation of AH x = λx, we have n X

ajt xj = λxt ,

j=1

(λ − att )xt =

n X

atj xj ,

t6=j=1

taking absolute values and applying the triangle inequality, we obtain |xt | |λ − att | ≤

5

n X

t6=j=1

|atj | |xj |,

since, |xt | ≥ |xi | ∀ i (1 ≤ i ≤ n), then |λ − att | ≤

n X

t6=j=1

|ajt | := Ct . 

Next we derive the localization theorem in terms of deleted absolute row and column sums of A and this is known as generalized Ostrowski theorem which is as follows. Theorem 3.4 (The generalized Ostrowski theorem for left eigenvalues) Let A := (aij ) ∈ Mn (H) and let γ ∈ [0, 1]. Then all the left eigenvalues of A are located in the union of n balls

Ti (A) := {z ∈ H : |z − aii | ≤ Riγ Ci1−γ }, i.e.,

Λl (A) ⊆ T (A) := ∪ni=1 Ti (A). Proof. Let λ be a left eigenvalue of A. Then by (Theorem 6, [22]), and for any γ ∈ [0, 1], |λ − aii |γ ≤ Riγ ,

(1)

similarly, from Theorem 3.3, we obtain |λ − aii |1−γ ≤ Ci1−γ .

(2)

Combining Equation (1) and Equation (2) |λ − aii | ≤ Riγ Ci1−γ . Thus, all the left eigenvalues of A are located in the union of n balls Ti (A).  Corollary 3.5 For any A := (aij ) ∈ Mn (H), n ≥ 2 and for any γ ∈ [0, 1], assume that |aii | > Riγ Ci1−γ , ∀ i (1 ≤ i ≤ n).

(3)

Then A is nonsingular. Proof. On the contrary, suppose that A is singular. Then from Theorem 2.3, there is a left eigenvalue λ = 0 of A. Then from Theorem 3.4, we obtain |aii | ≤ Riγ Ci1−γ , which contradicts Equation (3). Hence A is nonsingular. Next we extend the Brauer theorem from complex to quaternionic matrices for deleted absolute column sums. The quaternionic version of Brauer theorem for deleted absolute column sums of A can be found in the literature [12]. In this report it has been explained that if λ ∈ Λl (A) for A ∈ Mn (H), then its conjugate λ lies in the union of

n(n−1) 2

Cassini which has been found to be incorrect, follows from the following example. 6

ovals of



Example 3.6 Consider A := 

i

j

−j i



.

Then by [12, Theorem 5], oval of Cassini is given by {z ∈ H : |z − i| ≤ 1} . In this example, i − k is a left eigenvalue of A and its conjugate −i + k is not contained in the above oval of Cassini. Now we derive the corrected version of the Brauer theorem in terms of deleted absolute column sums of A as follows. Theorem 3.7 Let A := (aij ) ∈ Mn (H). Then all the left eigenvalues of A are located in the union of

n(n−1) 2

ovals of Cassini; Qij (A) := {z ∈ H : |z − aii | |z − ajj | ≤ Ci Cj } ,

i.e., Λl (A) ⊆ Q(A) := ∪ni,j=1, Qij (A). i6=j

Proof. Let λ be a left eigenvalue of A. Then by Theorem 3.2, λ is a left eigenvalue of AH . Then AH x = λx for some non-zero x := [x1 , . . . , xn ]T ∈ Hn . Suppose at least two non-zero entries of quaternion vector x are non-zero, say xt and xs . Assume |xs | := max {|xi |} and |xt | := max {|xi |}, i 6= s such that |xs | ≥ |xt | ≥ |xi | ∀ i (1 ≤ 1≤i≤n

1≤i≤n

i ≤ n), and xs 6= 0 6= xt . From s-th equation of AH x = λx, we have n X

ajs xj = λxs ,

j=1

i.e., (λ − ass )xs =

n X

ajs xj ,

s6=j=1

by the triangle inequality,

it is equivalent to

X n n X ajs xj ≤ |ajs | |xj |, |xs | |λ − ass | = s6=j=1 s6=j=1 |xs | |λ − ass | ≤

n X

s6=j=1

|ajs | |xt | = Cs |xt |,

this implies |λ − ass | ≤

 7

|xt | |xs |



Cs .

(4)

Similarly, from AH x = λx, we get |λ − att | ≤



|xs | |xt |



Ct .

(5)

Combining Equation (4) and Equation (5)     |xt | |xs | |λ − ass | |λ − att | ≤ Cs Ct = Cs Ct , |xs | |xt | i.e., |λ − ass | |λ − att | ≤ Cs Ct , (1 ≤ s, t ≤ n, s 6= t). Hence, all the right eigenvalues of A are located in the union of

n(n−1) 2

ovals of Cassini

Qij (A) (1 ≤ i, j ≤ n, i 6= j).  Corollary 3.8 Let A := (aij ) ∈ Mn (H) and |ass ||att | > Cs Ct , (1 ≤ s, t ≤ n, s 6= t). Then A is invertible. Proof. It is easy to prove by using Theorem 3.7, so we omit the proof.  [12, Theorem 7] has been stated for the central closed quaternionic matrix, now we generalize for all quaternionic matrices as follows. Theorem 3.9 Let A := (aij ) ∈ Mn (H) and let γ ∈ [0, 1]. Then all the left eigenvalues of A n(n − 1) are located in the union of ovals of Cassini; 2 o n Kij (A) := z ∈ H : |z − aii | |z − ajj | ≤ Riγ Rjγ Ci1−γ Cj1−γ , i.e., Λl (A) ⊆ K(A) := ∪ni,j=1 Kij (A). i6=j

Proof. Let λ be a left eigenvalue of A. Then by [12, Theorem 4], Theorem 3.7, and for any γ ∈ [0, 1], we obtain

|λ − aii |γ |λ − ajj |γ ≤ Riγ Rjγ ,

(6)

|λ − aii |1−γ |λ − ajj |1−γ ≤ Ci1−γ Cj1−γ .

(7)

and

From Equation (6) and Equation (7), we obtain |λ − aii | |λ − ajj | ≤ Riγ Rjγ Ci1−γ Cj1−γ . 8

Thus, all the left eigenvalues of A are located in the union of

n(n−1) 2

ovals of Cassini Kij (A). 

Thus Theorem 3.9 generalizes [12, Theorem 7]. Now, the following result shows that the Theorem 3.9 is sharper than the Theorem 3.4. Theorem 3.10 Let A := (aij ) ∈ Mn (H) with n ≥ 2 and let γ ∈ [0, 1] be given. Then K(A) ⊆ T (A). Proof. Let z ∈ Kij (A) and fix any i and j (1 ≤ i, j ≤ n, i 6= j), then from Theorem 3.9, we have γ γ 1−γ 1−γ |z − aii | |z − ajj | ≤ Ri Rj Ci Cj .

(8)

Then following two cases are possible, γ γ 1−γ 1−γ Case 1: If Ri Rj Ci Cj = 0, then z = aii or z = ajj . However from Theorem 3.4, we have aii ∈ Ti (A) and ajj ∈ Tj (A). Thus z ∈ Ti (A) ∪ Tj (A). γ γ 1−γ 1−γ Case 2: If Ri Rj Ci Cj > 0, then by Equation (8)  ! |z − aii |  |z − ajj |  ≤ 1. γ 1−γ γ 1−γ Ri C i Rj C j

(9)

As the left side of Equation (9) cannot exceed unity, then one of the factors of the left side is at most unity, i.e., z ∈ Ti (A) or z ∈ Tj (A). Hence z ∈ Ti (A) ∪ Tj (A), so Kij ⊆ Ti (A) ∪ Tj (A).

(10)

From Theorem 3.4 and Theorem 3.9, Equation (10) implies K(A) := ∪ni,j=1 Kij (A) ⊆ ∪ni,j=1 {Ti (A) ∪ Tj (A)} = ∪nk=1 Tk (A) =: T (A).  i6=j

i6=j

We need the following results for subsequent development of our theory. Theorem 3.11 [23]. Let A := (aij ) ∈ Mn (H) with aii ∈ R. Then all the right eigenvalues of A are located in the union of

n(n−1) 2

ovals of Cassini;

(a) Λr (A) ⊆ L(A) := ∪ni,j=1 {z ∈ H : |z − aii | |z − ajj | ≤ Ri Rj } . i6=j

(b) Λr (A) ⊆ Q(A) := ∪ni,j=1 {z ∈ H : |z − aii | |z − ajj | ≤ Ci Cj } . i6=j

9

Theorem 3.12 [23, Theorem 4.3]. Let A := (aij ) ∈ Mn (H) with aii ∈ R and let γ ∈ [0, 1] be given. Then all the right eigenvalues of A are located in the union of n Gerschgorin balls o n Gi (A) := z ∈ H : |z − aii | ≤ Riγ Ci1−γ , i.e., Λr (A) ⊆ G(A) := ∪ni=1 Gi (A). Now, we present the following result which is sharper than the Theorem 3.12. Theorem 3.13 Let A := (aij ) ∈ Mn (H) with aii ∈ R and γ ∈ [0, 1] be given. Then all n(n − 1) the right eigenvalues of A are located in the union of ovals of Cassini Gij (A) := 2 o n γ γ z ∈ H : |z − aii | |z − ajj | ≤ Ri Rj Ci 1−γ Cj 1−γ , i.e., Λr (A) ⊆ G(A) = ∪ni,j=1 Gij (A). i6=j

Proof. Let λ be a right eigenvalue of A. Then, by Theorem 3.11 and for any γ ∈ [0, 1], we have |λ − aii |γ |λ − ajj |γ ≤ Riγ Rjγ ,

(11)

|λ − aii |1−γ |λ − ajj |1−γ ≤ Ci1−γ Cj1−γ .

(12)

and

From Equation (11) and Equation (12) |λ − aii | |λ − ajj | ≤ Riγ Rjγ Ci1−γ Cj1−γ . Thus, the right eigenvalues of A ∈ Mn (H) are located in the union of

n(n − 1) ovals of Cassini 2

Gij (A) (1 ≤ i, j ≤ n, i 6= j).  We derive the following theorem which states that Theorem 3.13 is sharper than the Theorem 3.12. Theorem 3.14 Let A := (aij ) ∈ Mn (H), n ≥ 2 with aii ∈ R and let γ ∈ [0, 1] be given, then G(A) ⊆ G(A), where G(A) and G(A) are in Theorem 3.12 and Theorem 3.13, respectively. Proof. Let z ∈ Gij (A), (1 ≤ i, j ≤ n, i 6= j). For fix i and j, Theorem 3.13 gives |z − aii | |z − ajj | ≤ Riγ Rjγ Ci1−γ Cj1−γ . 10

(13)

Then following two cases are possible, Case 1 : If Riγ Rjγ Ci1−γ Cj1−γ = 0, then z = aii or z = ajj . However, from Theorem 3.12, we have aii ∈ Gi (A) and ajj ∈ Gj (A). Thus z ∈ Gi (A) ∪ Gj (A). Case 2 : If Riγ Rjγ Ci1−γ Cj1−γ > 0, then by Equation (13) ! ! |z − ajj | |z − aii | ≤ 1. Riγ Ci1−γ Rjγ Cj1−γ

(14)

As the left side of Equation (14) cannot exceed unity, then one of the factors of the left side is at most unity, i.e., z ∈ Gi (A) or z ∈ Gj (A). Hence Gij ⊆ Gi (A) ∪ Gj (A), i.e., G(A) := ∪ni,j=1 Gij (A) ⊆ ∪ni,j=1 {Gi (A) ∪ Gj (A)} = ∪nk=1 Gk (A) = G(A).  i6=j

i6=j

Now onwards we define 

for A := (aij ) ∈ Mn (H).

Ni := 

n X

i6=j=1

1

v uX u n p ′ |aij | , Ni := t |aij |2 p

i6=j=1

Now we discuss some different distribution theorems for left as well as for right eigenvalues of quaternionic matrix using H¨ older’s inequality are as follows. Theorem 3.15 Let A := (aij ) ∈ Mn (H) and let γ ∈ [0, 1] be given. Then all the left eigenvalue of A are contained in the union of n generalized balls o n 1−γ Bi (A) := z ∈ H : |z − aii | ≤ (n − 1) q Riγ Ni1−γ ,

i.e.,

Λl (A) ⊆ B(A) := ∪ni=1 Bi (A). Proof. Let µ be a left eigenvalue of A. Then Ax = µx for some non-zero vector x := [x1 , . . . , xn ]T ∈ Hn . Let xt be an element of x such that |xt | ≥ |xi | ∀ i (1 ≤ i ≤ n). Then |xt | > 0. Thus from Ax = µx, we have att xt +

n X

atj xj = µxt ,

t6=j=1

(µ − att )xt = 11

n X

t6=j=1

atj xj ,

X n X n |µ − att ||xt | = atj xj ≤ |atj | |xj |, t6=j=1 t6=j=1

(15)

applying generalized H¨ older’s inequality (Lemma 2.4) to Equation (15) 

|µ − att ||xt | ≤ 

1  p

n X

t6=j=1

p

|atj |

since |xt | ≥ |xi | ∀ i (1 ≤ i ≤ n), we conclude



1 q

n X

t6=j=1

q

|xj |

,

1

|µ − att ||xt | ≤ Nt ((n − 1)|xt |q ) q , i.e., 1

|µ − att | ≤ Nt (n − 1) q .

(16)

From Equation (15) and by using |xt | ≥ |xi | ∀ i (1 ≤ i ≤ n) |µ − att ||xt | ≤ i.e., |µ − att | ≤

n X

t6=j=1

n X

t6=j=1

|atj | |xt |,

|atj | = Rt .

(17)

Therefore, for any γ ∈ [0, 1], and from Equation (16), Equation (17) |µ − att |1−γ ≤ Nt1−γ (n − 1)

1−γ q

and |µ − att |γ ≤ Rtγ ,

(18)

i.e., |µ − att | ≤ (n − 1)

1−γ q

Nt1−γ Rtγ .

Thus theorem is proved.  Following example is now given to illustrate the above result.   −i − j 1 − 2k  . From Theorem 3.15 for p = q = Example 3.16 Consider a matrix A :=  1 −i + j 2, γ = 1/2, we obtain B(A) := {z ∈ H : |z + i + j| ≤ 2.2361} ∪ {z ∈ H : |z + i − j| ≤ 1}. Hence the above Example illustrates the Theorem 3.15 and hence it is verified. Now for the special cases we find the results available in the literatures. 12

Corollary 3.17 Let A := (aij ) ∈ Mn (H) and let γ ∈ [0, 1] be given. Then all the left eigenvalue of A are contained in the union of n balls; n o 1−γ (1−γ) Di (A) := z ∈ H : |z − aii | ≤ (n − 1) 2 Riγ Ni′ , i.e., Λl (A) ⊆ D(A) := ∪ni=1 Di (A). Proof. Assume p = q = 2 in the previous Theorem, we obtain the required result. This can be found in [5, Theorem 1].  Corollary 3.18 Let A := (aij ) ∈ Mn (H). Then all the left eigenvalue of A are contained in the union of n balls Γi (A) := {z ∈ H : |z − aii | ≤ Ri } , i.e., Λl (A) ⊆ Γ(A) := ∪ni=1 Γi (A). Proof. Assume p = q = 2, γ = 1, in the previous Theorem, we obtain the required result. This can be found in [22, Theorem 6].  Corollary 3.19 Let A := (aij ) ∈ Mn (H). Then all the left eigenvalues of A are contained in o n 1 the union of n balls Si (A) := z ∈ H : |z − aii | ≤ (n − 1) 2 Ni′ , i.e., Λl (A) ⊆ S(A) := ∪ni=1 Si (A). Proof. Assume p = q = 2, γ = 0, in the previous Theorem, we obtain the required result. This can be found in [20, Theorem 1].  Now we present the generalization of the [22, Theorem 7] and [23, Theorem 3.1]. In case of general quaternionic matrix, all right eigenvalues may not lie in generalized balls Bi (A), however we show that every connected region of generalized balls Bi (A) contains some right eigenvalues of A. Theorem 3.20 Let A := (aij ) ∈ Mn (H) and let γ ∈ [0, 1] be given. For every right eigenvalue µ of A there exists a quaternion β such that β −1 µβ ( which is also a right eigenvalue ) is contained in the union of n generalized balls; o n 1−γ Bi (A) := z ∈ H : |z − aii | ≤ (n − 1) q Riγ Ni 1−γ ,  i.e., β −1 µβ : 0 6= β ∈ H ∩ ∪ni=1 Bi (A) 6= ∅.

13

Proof.

Let µ be a right eigenvalue of A. Then there exists a non-zero vector x :=

[x1 , . . . , xn ]T ∈ Hn such that Ax = xµ. Let xt be an element of x such that |xt | ≥ |xi | ∀ i (1 ≤ i ≤ n). Then |xt | > 0. Consider ρ such that xt µ = ρxt . Then ρ is similar to µ and thus we have the following att xt +

n X

atj xj = xt µ

t6=j=1 n X

(ρ − att )xt =

atj xj

j=1,j6=t

X n X n |ρ − att ||xt | = atj xj ≤ |atj | |xj |. t6=j=1 j=1,j6=t

(19)

By generalized H¨ older’s inequality (Lemma 2.4), we have 

|ρ − att ||xt | ≤ 

1  p

n X

t6=j=1

p

|atj |

Since |xt | ≥ |xi | ∀ i (1 ≤ i ≤ n), we conclude that



n X

t6=j=1

1 q

q

|xj |

.

1

|ρ − att ||xt | ≤ Nt ((n − 1)|xt |q ) q , i.e., 1

|ρ − att | ≤ Nt (n − 1) q .

(20)

Further from Equation (19) with |xt | ≥ |xi | ∀ i (1 ≤ i ≤ n) |ρ − att ||xt | ≤ i.e., |ρ − att | ≤

n X

t6=j=1

n X

t6=j=1

|atj | |xt |,

|atj | = Rt .

(21)

Combining Equation (20) and Equation (21) |ρ − att |1−γ ≤ Nt1−γ (n − 1)

1−γ q

and |ρ − att |γ ≤ Rtγ ,

i.e., |ρ − att | ≤ (n − 1)

1−γ q

Nt1−γ Rtγ .

Thus theorem is proved.  Now we see the following results are available in the literatures for special cases. 14

(22)

Corollary 3.21 Let A := (aij ) ∈ Mn (H). For every right eigenvalue λ of A there exists a quaternion α such that α−1 λα ( which is also a right eigenvalue ) is contained in the union of n Gerschgorin balls {z ∈ H : |z − aii | ≤ Ri }, i.e., {α−1 λα : 0 6= α ∈ H} ∩ ∪ni=1 {z ∈ H : |z − aii | ≤ Ri } = 6 ∅. Proof. Substituting p = q = 2, γ = 1, in the previous Theorem, we obtain the required result. This can be found in [22, Theorem 7].  Corollary 3.22 Let A := (aij ) ∈ Mn (H). For every right eigenvalue λ of A there exists a quaternion α such that α−1 λα (which is also a right eigenvalue) is contained in the union of √ n Gerschgorin balls {z ∈ H : |z − aii | ≤ n − 1 Ni′ }, i.e., √  {α−1 λα : 0 6= α ∈ H} ∩ ∪ni=1 z ∈ H : |z − aii | ≤ n − 1 Ni′ 6= ∅.

Proof. Substituting p = q = 2, γ = 0, in the previous Theorem, we obtain the required result. This can be found in [23, Theorem 3.1].  Now we derive the following theorems when the diagonals of square matrix A ∈ Mn (H) are real. Theorem 3.23 Let A := (aij ) ∈ Mn (H) with aii ∈ R and let γ ∈ [0, 1] be given. Then all the right eigenvalues of A are in the union of n generalized balls n o 1−γ Bi (A) := z ∈ H : |z − aii | ≤ (n − 1) q Ri γ Ni 1−γ , i.e., Λr (A) ⊆ B(A) := ∪ni=1 Bi (A). Proof. Proof is immediate from Theorem 3.15, so we skip the proof.  Following example is now given to illustrate our result.   1 j k     Example 3.24 Let A := 1 + i 2 i + j .   1−j j+k 4 From Theorem 3.23 for p = q = 2, γ = 1/2, we obtain B(A) := {z ∈ H : |z − 1| ≤ 2} ∪ {z ∈ H : |z − 2| ≤ 2.8284} ∪ {z ∈ H : |z − 4| ≤ 2.8284}. Thus the right eigenvalues of A are contained in B(A). Thus the Theorem 3.23 is valid. 15

4

Bounds for the absolute sum of left and right eigenvalues

In this section, consider A := (aij ) ∈ Mn (H) and define n X

Υ1 :=

i=1

n X

Υ2 :=

i=1

where

1 p

+

1 q

|aii | + (n − 1) |aii | + (n − 1)

1−γ q

n X

Riγ Ni1−γ

i=1

1−γ q

n X

Riγ Ni1−γ

 n  X trace(A) + aii − + |trace(A)| , n i=1

i=1

= 1 and p, q ∈ (1, ∞), γ ∈ [0, 1]. Now we derive bounds for the absolute sum of

the left and right eigenvalue of quaternionic matrix. Theorem 4.1 Let A := (aij ) ∈ Mn (H) and let γ ∈ [0, 1] be given. Suppose λi (1 ≤ i ≤ n), are distinct left eigenvalues of A which lie in n distinct generalized balls Bi (A), respectively, then (a) (b)

Pn

i=1 |λi |

Pn

i=1 |λi |

≤ Υ1 , ≤ Υ2 .

Proof. (a) Consider λi are n distinct left eigenvalues of A which lie in n distinct generalized balls Bi (A). Then, without loss of generality, we consider λi ∈ Bi (A) := {z ∈ H : |z − aii | ≤ (n − 1)

1−γ q

Riγ Ni1−γ }, where Bi (A) 6= Bj (A) (1 ≤ i, j ≤ n, i 6= j).

Then by Theorem 3.15, we have |λi − aii | ≤ (n − 1) |aii | + (n − 1)

1−γ q

Riγ Ni1−γ . Then, n X i=1

|λi | ≤

n X i=1

|aii | + (n − 1)

1−γ q

1−γ q

Riγ Ni1−γ . Hence, we obtain |λi | ≤

n X

Riγ Ni1−γ .

i=1

(b) Consider λi are n distinct left eigenvalues of A which lie within n distinct generalized balls Bi (A). Then, without loss of generality, we consider λi ∈ Bi (A) = {z ∈ H : |z − aii | ≤ (n − 1)

1−γ q

Riγ Ni1−γ }, where Bi (A) 6= Bj (A) (1 ≤ i, j ≤ n, i 6= j). Based on the particle and

center gravity theorem, each Bi (A) can be treated as a particle or a rigid body. Then the P . Now, we have center of all particles or rigid bodies is n1 ni=1 aii = trace(A) n λi − trace(A) = λi − aii + aii − trace(A) ≤ |λi − aii | + aii − trace(A) n n n 1−γ λi − trace(A) ≤ (n − 1) q Rγ N 1−γ + aii − trace(A) . i i n n 16

This implies |λi | ≤ (n − 1) n X i=1

|λi | ≤ (n − 1)

1−γ q

1−γ q

Riγ Ni1−γ + aii −

n X

Riγ Ni1−γ (A)



trace(A) n

+ trace(A) , and n

 n  X trace(A) + aii − + |trace(A)| .  n i=1

i=1

Now, suppose γ = 1 in the previous Theorem 4.2(a), we obtain n X i=1

|λi | ≤

n X n X i=1 j=1

|aij |.

This has been presented in [13, Theorem 3.1]. Assume p = q = 2, γ = 0 in the previous Theorem 4.2(a), we obtain v  uX n n n X X X p u n t |aii |. |aij |2  + |λi | ≤ (n − 1) i6=j=1

i=1

i=1

i=1

This can be found in [13, Theorem 3.2].

Consider γ = 1 in the previous Theorem 4.2(b), we obtain  n n  n X n X X X trace(A) |aij | + |λi | ≤ aii − + |trace(A)| . n i=1

i=1 i6=j=1

i=1

This has been discussed in [13, Theorem 3.3]. 

Now we show that the above bounds are same for the right eigenvalues of quaternionic matrix as well, which are as follows Theorem 4.2 Let A := (aij ) ∈ Mn (H) and let γ ∈ [0, 1] be given. Suppose ρ−1 i λi ρi , where λi ∈ Λr (A) and 0 6= ρi ∈ H, (1 ≤ i ≤ n), are right eigenvalues of A that lie in n distinct generalized balls Bi (A), then (a) (b)

Pn

i=1 |λi |

Pn

i=1 |λi |

≤ Υ1 , ≤ Υ2 .

Proof. (a) Consider ρ−1 i λi ρi are n right eigenvalues of A which lie in n distinct generalized balls Bi (A). Then, without loss of generality, we consider ρ−1 i λi ρi ∈ Bi (A) := {z ∈ H : |z − aii | ≤ (n − 1)

1−γ q

Riγ Ni1−γ }, where Bi (A) 6= Bj (A), 1 ≤ i, j ≤ n, i 6= j.

Then by Theorem 3.20, we have |ρ−1 i λi ρi − aii | ≤ (n − 1)

|ρ−1 i λi ρi | = |λi | ≤ |aii | + (n − 1) n X i=1

1−γ q

|λi | ≤

1−γ q

Riγ Ni1−γ . Hence, we obtain

Riγ Ni1−γ . Then,

n X i=1

|aii | + (n − 1) 17

1−γ q

n X i=1

Riγ Ni1−γ . 

(b) Consider ρ−1 i λi ρi are n right eigenvalues of A which lie within n distinct generalized balls Bi (A). Then, without loss of generality, we consider ρ−1 i λi ρi ∈ Bi (A) = {z ∈ H : |z − aii | ≤ (n − 1)

1−γ q

Riγ Ni1−γ }, where Bi (A) 6= Bj (A), 1 ≤ i, j ≤ n, i 6= j. Based on the

particle and center gravity theorem, each Bi (A) can be treated as a particle or a rigid body. n 1X trace(A) Then the center of all particles or rigid bodies is aii = . Now, we have n n i=1

−1 −1 −1 trace(A) trace(A) trace(A) ρ λi ρi − = ρ λi ρi − aii + aii − ≤ ρ λi ρi − aii + aii − i i i n n n −1 1−γ trace(A) trace(A) γ 1−γ ≤ (n − 1) q R N . ρ λi ρi − + aii − i i i n n 1−γ trace(A) trace(A) γ 1−γ q R N a − + + Hence |ρ−1 λ ρ | = |λ | ≤ (n − 1) . Thus, we have ii i i i i i i n n n X i=1

|λi | ≤ (n − 1)

1−γ q

n X

Riγ Ni1−γ (A)

 n  X trace(A) + aii − + |trace(A)| .  n i=1

i=1

Now we have the following table to verify the above bounds for γ = 1/2, p = q = 2. Example 4.3 

i+j

j

  A :=  0  0 Matrix: A λi ∈ Λl (A) λi ∈ Λr (A)

5

k/2



  j/2   0 −j + k

k

P3

i=1 |λi |

Υ1

Υ2

3.8284

5.9630

11.1504

3.8284

5.9630

11.1504

Conclusions

In this paper, we have derived Ostrowski type theorem for left eigenvalues of quaternionic matrix. The corrected version of the Brauer type theorem for left eigenvalues in terms of deleted absolute column sums is given. Further, we have discussed the sufficient conditions for a matrix A ∈ Mn (H) to be nonsingular. In addition, the generalizations of the Gerschgorin type theorems and some other localization theorems have been discussed for left as well as for right eigenvalues. Sharper results than Ostrowski type theorems have been presented. Finally, we have derived the generalizations of the bounds for the absolute sum of left and right eigenvalues of a quaternionic matrix. 18

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