Bounds for the right spectral radius of quaternionic matrices
UDC 517.5  In this paper  we present bounds for the sum of the moduli of right eigenvalues of a quaternionic matrix. As a consequence, we obtain bounds for the right spectral radius of a quaternionic matrix. We also present a minimal ball in 4D spaces which contains all the...
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Institute of Mathematics, NAS of Ukraine
2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512219743125504 |
|---|---|
| author | Ali, I. Ali, I. |
| author_facet | Ali, I. Ali, I. |
| author_sort | Ali, I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:01:50Z |
| description | UDC 517.5 
In this paper  we present bounds for the sum of the moduli of right eigenvalues of a quaternionic matrix. As a consequence, we obtain bounds for the right spectral radius of a quaternionic matrix. We also present a minimal ball in 4D spaces which contains all the Gersgorin balls of a quaternionic matrix. As an application, we introduce the estimation for the right ˇ eigenvalues of quaternionic matrices in the minimal ball. Finally, we suggest some numerical examples to illustrate of our results. |
| doi_str_mv | 10.37863/umzh.v72i6.6018 |
| first_indexed | 2026-03-24T03:25:19Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i6.6018
UDC 517.5
I. Ali (School Basic Sci., Indian Inst. Technology Indore, Simrol, India)
BOUNDS FOR THE RIGHT SPECTRAL RADIUS
OF QUATERNIONIC MATRICES*
ГРАНИЧНI ОЦIНКИ ДЛЯ ПРАВОГО СПЕКТРАЛЬНОГО РАДIУСА
МАТРИЦЬ КВАТЕРНIОНIВ
In this paper, we present bounds for the sum of the moduli of right eigenvalues of a quaternionic matrix. As a consequence,
we obtain bounds for the right spectral radius of a quaternionic matrix. We also present a minimal ball in 4D spaces
which contains all the Geršgorin balls of a quaternionic matrix. As an application, we introduce the estimation for the right
eigenvalues of quaternionic matrices in the minimal ball. Finally, we suggest some numerical examples to illustrate of our
results.
Знайдено граничнi оцiнки для сум модулiв правих власних значень кватернiонних матриць. Як наслiдок отримано
оцiнки для правого спектрального радiуса таких матриць. У чотиривимiрних просторах знайдено мiнiмальний шар,
який мiстить всi шари Гершгорiна матрицi кватернiонiв. Як застосування запропоновано оцiнку для правих власних
значень матриць кватернiонiв. Також наведено приклади для iлюстрацiї цих результатiв.
1. Introduction. The problems over a quaternion division algebra have received much attention in
the literature due to their applications in pure and applied sciences, such as the quantum physics,
control theory, altitude control, computer graphics and signal processing (see, for example, [1, 2,
4 – 6, 12, 14, 20 – 22] and the references therein). There are many research paper published on the
location and estimation of the left and right eigenvalues of a quaternionic matrix [8, 16, 20, 22,
23]. The stability of linear difference/differential equations with quaternionic matrix coefficients is
based on the location of right eigenvalues of their corresponding quaternionic block matrices [10, 11,
15]. The upper bound for the left and right spectral radius of a quaternionic matrix has proposed by
F. Zhang [22] in terms of the operator norm of a quaternionic matrix. Bounds for the sum of the
left eigenvalues norms are derived with the help of localization theorems for left eigenvalues of a
quaternionic matrix [8]. The first attempts to locate the zeros of quaternionic polynomials were given
by G. Opfer [9] by direct calculation.
In the first part of the paper, we present bounds for the sum of the absolute values of right
eigenvalues of a quaternionic matrix. We further discuss bounds for the right spectral radius of
a quaternionic matrix by applying the above theory. In the second part of the paper, we provide
a minimal ball which contains all the Geršgorin balls of a quaternionic matrix. Then we give
localization theorems for right eigenvalues of a quaternionic matrix with the help of the above
minimal ball.
The paper is organized as follows. Section 2 reviews some existing results. Section 3 discusses
upper bounds for the sum of the right eigenvalues norms and the right spectral radius of a quaternionic
* This research was supported by the CSIR, Govt. of India.
c\bigcirc I. ALI, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6 723
724 I. ALI
matrix. Finally, Section 4 presents a minimal ball and location for right eigenvalues of a quaternionic
matrix.
2. Preliminaries. Throughout the paper, \BbbR and \BbbC denote the fields of real and complex numbers,
respectively. The set of real quaternions is defined by
\BbbH := \{ q = q0 + q1\bfi + q2\bfj + q3\bfk : q0, q1, q2, q3 \in \BbbR \}
with \bfi 2 = \bfj 2 = \bfk 2 = \bfi \bfj \bfk = - 1. This relation implies that \bfi \bfj = - \bfj \bfi = \bfk , \bfj \bfk = - \bfk \bfj = \bfi ,
\bfk \bfi = - \bfi \bfk = \bfj . The conjugate of q \in \BbbH is q := q0 - q1\bfi - q2\bfj - q3\bfk and the modulus of q is
| q| :=
\sqrt{}
q20 + q21 + q22 + q23. \Im (a) denotes the imaginary part of a \in \BbbC . The real part of a quaternion
q = q0 + q1\bfi + q2\bfj + q3\bfk is defined as \Re (q) = q0. Let p, q \in \BbbH . Then (a) | q| = | q| and pq = q p;
(b) | pq| = | qp| = | p| | q| ; (c) \bfj c = c\bfj or \bfj c\bfj = c for every c \in \BbbC ; (d) p - 1 =
p
| p| 2
if p \not = 0, and
| \rho - 1p\rho | = | p| for all \rho \in \BbbH \setminus \{ 0\} .
The collection of all n-column vectors with elements in \BbbH is denoted by \BbbH n. For x \in \scrK n, where
\scrK \in \{ \BbbR ,\BbbC ,\BbbH \} , the transpose of x is xT . If x = [x1, . . . , xn]
T , then the conjugate of x is defined as
x = [x1, . . . , xn]
T and the conjugate transpose of x is defined as xH = [x1, . . . , xn]. For x, y \in \BbbH n,
the inner product is defined as \langle x, y\rangle := yHx and the norm of x is defined as \| x\| :=
\sqrt{}
\langle x, x\rangle . The
sets of m \times n real, complex, and quaternionic matrices are denoted by Mm\times n(\BbbR ), Mm\times n(\BbbC ), and
Mm\times n(\BbbH ), respectively. When m = n, these sets are denoted by Mn(\scrK ), \scrK \in \{ \BbbR ,\BbbC ,\BbbH \} .
For A \in Mm\times n(\scrK ), the conjugate, transpose, and conjugate transpose of A are defined as
A = (aij) \in Mm\times n(\scrK ), AT = (aji) \in Mn\times m(\scrK ), and AH = (A)T \in Mn\times m(\scrK ), respectively. A
square matrix A \in Mn(\BbbH ) is said to be Hermitian if AH = A. We define the Frobenius norm on
A \in Mn(\BbbH ) by
\| A\| F :=
\bigl(
\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}AHA
\bigr) 1/2
.
Let p, q \in \BbbH . Then p and q are said to be similar, denoted by p \sim q, if
p \sim q \leftrightarrow \exists 0 \not = r \in \BbbH such that p = r - 1qr. (1)
The set
[p] := \{ u \in \BbbH : u = \rho - 1 p\rho for all 0 \not = \rho \in \BbbH \} (2)
is called an equivalence class of p \in \BbbH .
For any quaternionic matrix A = B1+B2\bfi +B3\bfj +B4\bfk \in Mn(\BbbH ), Bk \in Mn(\BbbR ), k = 1, 2, 3, 4,
A can be uniquely expressed as A = (B1 + B2\bfi ) + (B3 + B4\bfi )\bfj = A1 + A2\bfj , A1, A2 \in Mn(\BbbC ).
Define a function \Psi : Mn(\BbbH ) \rightarrow M2n(\BbbC ) by
\Psi A :=
\Biggl[
A1 A2
- A2 A1
\Biggr]
.
The matrix \Psi A is called the complex adjoint matrix of A. Unlike the complex matrix, there are two
types of eigenvalues of a quaternionic matrix, namely left and right.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
BOUNDS FOR THE RIGHT SPECTRAL RADIUS OF QUATERNIONIC MATRICES 725
Definition 2.1. Let A \in Mn(\BbbH ). Then the left, right, and the standard right eigenvalues, re-
spectively, are given by
\Lambda l(A) := \{ \lambda \in \BbbH : Ax = \lambda x for some nonzero x \in \BbbH n\} ,
\Lambda r(A) := \{ \lambda \in \BbbH : Ax = x\lambda for some nonzero x \in \BbbH n\}
and
\Lambda s(A) := \{ \lambda \in \BbbC : Ax = x\lambda for some nonzero x \in \BbbH n, \Im (\lambda ) \geq 0\} .
Definition 2.2. Let A \in Mn(\BbbH ). Then the right spectral radius of A is defined as
\rho r(A) := \mathrm{m}\mathrm{a}\mathrm{x} \{ | \lambda | : \lambda \in \Lambda r(A)\} .
Definition 2.3. Let A \in Mn(\BbbH ). Then A is said to be \eta -Hermitian if A = (A\eta )H , where
A\eta = \eta HA\eta and \eta \in \{ \bfi , \bfj ,\bfk \} .
Definition 2.4. A matrix A \in Mn(\BbbH ) is said to be invertible if there exists B \in Mn(\BbbH ) such
that AB = BA = In, where In is the n\times n identity matrix.
Definition 2.5. Let A \in Mn(\BbbH ). Then A is said to be a central closed matrix if there exists an
invertible matrix T such that T - 1AT = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mu 1, \mu 2, . . . , \mu n), where \mu i \in \BbbR , 1 \leq i \leq n.
We recall the following results for the development of our theory.
Lemma 2.1 [19]. Let A \in Mn(\BbbH ) be a central closed matrix and suppose that the standard
right eigenvalues of A are \mu 1, \mu 2, . . . , \mu n. Then \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A) =
\sum n
i=1
\mu i.
Theorem 2.1 ([13], Theorem 3.1). Let A = (aij) \in Mn(\BbbH ) be a central closed matrix. Then
all the standard right eigenvalues of A are located in the following ball:
G(A) =
\biggl\{
z \in \BbbH :
\bigm| \bigm| \bigm| \bigm| z - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \leq \xi (A)
\biggr\}
,
where
\xi (A) =
\sqrt{}
n - 1
2n - 1
\sqrt{}
n - 1
n
\eta +
\sqrt{}
\eta 2 - 2n - 1
n2
F (A),
\eta =
\biggl(
\| A\| 2F -
\bigm| \bigm| \bigm| \bigm| \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| 2\biggr) , F (A) = \| AAH\| 2F - \| A2\| 2F .
Let A := (aij) \in Mn(\BbbH ). Then define the deleted absolute row sums of A as
ri(A) :=
n\sum
j=1, j \not =i
| aij | , 1 \leq i \leq n.
We also define the n Geršgorin balls as follows:
Gi(A) := \{ z \in \BbbH : | z - aii| \leq ri(A)\} , 1 \leq i \leq n.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
726 I. ALI
3. Bounds for the sum of the right eigenvalue norms for a quaternionic matrix. In view of
Definition 2.2, one can compute the right spectral radius of a quaternionic matrix by means of the
following simple procedure.
Let A \in Mn(\BbbH ).
Factorize A = A1 +A2\bfj , where A1, A2 \in Mn(\BbbC ).
Write the complex adjoint matrix \Psi A :=
\Biggl[
A1 A2
- A2 A1
\Biggr]
.
Find \Lambda (\Psi A) = \{ \mu 1, . . . , \mu n, \mu 1, . . . , \mu n\} .
Divide \Lambda (\Psi A) into two sets \Lambda 1, \Lambda 2 such that \Lambda 1 \cup \Lambda 2 = \Lambda (\Psi A), the eigenvalues with po-
sitive imaginary part belong to \Lambda 1 and the elements of \Lambda 2 are the conjugates of the ones of \Lambda 1.
Consequently, from Definition 2.1, \Lambda 1 is the set of the standard right eigenvalues of A.
Then the right spectral radius of A is given as
\rho r(A) := \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\{ | \mu i| : \mu i \in \Lambda s(A) := \Lambda 1\} . (3)
One can also compute all the right eigenvalues of a quaternionic matrix with the help of standard
right eigenvalues of that matrix. In view of the above points, we can obtain the right spectrum of A
as follows:
\Lambda r(A) =
n\bigcup
i=1
[\mu i], \mu i \in \Lambda s(A) := \Lambda 1.
The Geršgorin theorem for right eigenvalues of a quaternionic matrix has proved by F. Zhang
[22] which is as follows.
Lemma 3.1 ([22], Theorem 7). Let A := (aij) \in Mn(\BbbH ). For every right eigenvalue \mu of A
there exists a nonzero quaternion \beta such that \beta - 1\mu \beta (which is also a right eigenvalue) is contained
in the union of n Geršgorin balls Gi(A) := \{ z \in \BbbH : | z - aii| \leq ri(A)\} , 1 \leq i \leq n, that is,
\bigl\{
z - 1\mu z : 0 \not = z \in \BbbH
\bigr\}
\cap
\Biggl(
n\bigcup
i=1
Gi(A)
\Biggr)
\not = \varnothing .
In particular, when \mu is real, it is contained in a Geršgorin ball.
First, in this section, we derive bounds for the sum of the absolute values of right eigenvalue of
a quaternionic matrix with the help of Theorem 3.1 which are as follows.
Theorem 3.1. Let A := (aij) \in Mn(\BbbH ). If \lambda i (1 \leq i \leq n) are right eigenvalues of A such that
they lie within n distinct Geršgorin balls Gi(A), respectively, then we have the following inequalities:
n\sum
i=1
| \lambda i| \leq
n\sum
i=1
n\sum
j=1
| aij | , (4)
n\sum
i=1
| \lambda i| \leq
n\sum
i=1
n\sum
j=1,j \not =i
| aij | +
n\sum
i=1
\biggl( \bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr) + | \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)| . (5)
Moreover, if \mu i, 1 \leq i \leq n, are standard right eigenvalues of A, then we have the following
inequalities:
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
BOUNDS FOR THE RIGHT SPECTRAL RADIUS OF QUATERNIONIC MATRICES 727
n\sum
i=1
| \mu i| \leq
n\sum
i=1
n\sum
j=1
| aij | , (6)
n\sum
i=1
| \mu i| \leq
n\sum
i=1
n\sum
j=1,j \not =i
| aij | +
n\sum
i=1
\biggl( \bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr) + | \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)| . (7)
Proof. Inequality (4): Since \lambda i, 1 \leq i \leq n, are n right eigenvalues of A such that they lie
within n distinct Geršgorin balls Gi(A) = \{ z \in \BbbH : | z - aii| \leq ri(A)\} , respectively. Now, without
loss of generality, we can consider as follows:
\lambda i \in Gi(A) and Gi(A) \not = Gj(A), 1 \leq i, j \leq n, i \not = j.
By applying Theorem 3.1, we have
| \lambda i - aii| \leq
n\sum
j=1,j \not =i
| aij | , 1 \leq i \leq n.
This implies that
| \lambda i| \leq | aii| +
n\sum
j=1,j \not =i
| aij | =
n\sum
j=1
| aij | .
Therefore,
n\sum
i=1
| \lambda i| \leq
n\sum
i=1
n\sum
j=1
| aij | .
Inequality (5): The Geršgorin balls Gi(A) = \{ z \in \BbbH : | z - aii| \leq ri(A)\} , 1 \leq i \leq n, have the
centres aii, respectively. Based on the particle and centre gravity theorem, each Gi(A), 1 \leq i \leq n,
can be considered as a particle or a rigid body. Then the centre of all particle or rigid body is
1
n
n\sum
i=1
aii =
\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
.
Now, we get\bigm| \bigm| \bigm| \bigm| \lambda i -
\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| \lambda i - aii + aii -
\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \leq | \lambda i - aii| +
\bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| ,\bigm| \bigm| \bigm| \bigm| \lambda i -
\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \leq ri(A) +
\bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| .
This implies that
| \lambda i| \leq ri(A) +
\bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| .
Therefore, we obtain
n\sum
i=1
| \lambda i| \leq
n\sum
i=1
ri(A) +
n\sum
i=1
\biggl( \bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr) +
n\sum
i=1
\bigm| \bigm| \bigm| \bigm| \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)n
\bigm| \bigm| \bigm| \bigm| .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
728 I. ALI
Thus, we have the following desired result:
n\sum
i=1
| \lambda i| \leq
n\sum
i=1
n\sum
j=1,j \not =i
| aij | +
n\sum
i=1
\biggl( \bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr) + | \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)| .
Inequality (6): Since \lambda i, 1 \leq i \leq n, are right eigenvalues of A, so from Lemma 3 of [3] there
exist \rho i \in \BbbH \setminus \{ 0\} , 1 \leq i \leq n, such that \rho - 1
i \lambda i\rho i = \mu i, where \mu , 1 \leq i \leq n, are the standard right
eigenvalues of A. This implies that \lambda i = \rho i\mu i\rho
- 1
i . From the inequality (4), we get
n\sum
i=1
| \rho i\mu i\rho
- 1
i | \leq
n\sum
i=1
n\sum
j=1
| aij | .
Since | \rho i\mu i\rho
- 1
i | = | \mu i| , 1 \leq i \leq n. Therefore,
\sum n
i=1
| \mu i| \leq
\sum n
i=1
\sum n
j=1
| aij | .
Similarly, from the inequality (5), we have the desired inequality (7), that is,
n\sum
i=1
| \mu i| \leq
n\sum
i=1
n\sum
j=1,j \not =i
| aij | +
n\sum
i=1
\biggl( \bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr) + | \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)| .
Theorem 3.1 is proved.
Now, from Theorem 3.1, we make the following observations for upper bounds of the right
spectral radius:
From (3), we can easily see that \rho r(A) \leq
\sum n
i=1
| \mu i| , \mu i \in \Lambda s(A). By applying inequality (6),
we have upper bound of the right spectral radius of A which is as follows:
\rho r(A) \leq
n\sum
i=1
n\sum
j=1
| aij | .
By applying inequality (7), we obtain upper bound of the right spectral radius of A in terms of
the \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e} of A which is as follows:
\rho r(A) \leq
n\sum
i=1
n\sum
j=1,j \not =i
| aij | +
n\sum
i=1
\biggl( \bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr) + | \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)| .
The upper bound of the left and right spectral radius of a quaternionic matrix has derived in [22]
in terms of the spectral norm of the quaternionic matrix. However, the spectral norm of a quaternionic
matrix is expensive to compute. Here, our bounds are in terms of moduli of entries and \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e} of a
quaternionic matrix which are much easier to compute than the spectral norm.
W. Junliang and Z. Yan [8] have been given Schur’s inequality for right eigenvalues of a quater-
nionic matrix which is as follows.
Lemma 3.2 ([8], Corollary 2.1). Let A := (aij) \in Mn(\BbbH ) and \mu 1, \mu 2, . . . , \mu n be standard right
eigenvalues of A. Then we have the following inequality:
n\sum
i=1
| \mu i| 2 \leq \| A\| 2F :=
n\sum
i=1
n\sum
j=1
| aij | 2. (8)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
BOUNDS FOR THE RIGHT SPECTRAL RADIUS OF QUATERNIONIC MATRICES 729
From Lemma 3.2, it is clear that the inequalities (6) and (7) are different from the inequality (8).
As applications of Theorem 3.1 and Lemma 3.2, we derive sharper estimations of the right spectral
radius in terms of the \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e} and the Frobenius norm.
Let \mu m be a standard right eigenvalue of maximum modulus. Then, from inequality (6), we can
be written as
| \mu m| \leq
n\sum
i=1
n\sum
j=1
| aij | -
n\sum
i \not =m
| \mu i| .
By applying the arithmetric-geometric mean inequality, we obtain
| \mu m| \leq
n\sum
i=1
n\sum
j=1
| aij | - (n - 1)
n\prod
i \not =m
| \mu i|
1
n - 1 \leq
\leq
n\sum
i=1
n\sum
j=1
| aij | - (n - 1)
\prod n
i=1
| \mu i|
1
n - 1
| \mu m|
1
n - 1
.
From [21] (Theorem 8.1(4)), we have
| \mu m| \leq
n\sum
i=1
n\sum
j=1
| aij | - (n - 1)
(| A| q)
1
2n - 2\Bigl( \sum n
i=1
\sum n
j=1
| aij |
\Bigr) 1
n - 1
,
where | A| q is the q-determinant of the quaternionic matrix A. The definition of the right spectral
radius gives
\rho r(A) \leq
n\sum
i=1
n\sum
j=1
| aij | - (n - 1)
(| A| q)
1
2n - 2\Bigl( \sum n
i=1
\sum n
j=1
| aij |
\Bigr) 1
n - 1
. (9)
Similarly, inequality (7) yields
\rho r(A) \leq \xi 1 - (n - 1)
(| A| q)
1
2n - 2
(\xi 1)
1
n - 1
, (10)
where \xi 1 =
\sum n
i=1
\sum n
j=1,j \not =i
| aij | +
\sum n
i=1
\biggl( \bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr) + | \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)| .
From the above procedure, Lemma 3.2 yields
\rho r(A) \leq
\Biggl[
\| A\| 2F - (n - 1)
\biggl(
| A| q
\| A\| 2F
\biggr) 1
n - 1
\Biggr] 1/2
. (11)
Let A \in Mn(\BbbH ). Then the discrete-time quaternionic system w(t + 1) = Aw(t) is said to be
asymptotically stable if and only if \Lambda r(A) \subset S\BbbH = \{ q \in \BbbH : | q| < 1\} . We present application of
bounds of the right spectral radius for the stability of a discrete-time quaternionic system. From the
above definition, if \rho r(A) < 1, then the system w(t+ 1) = Aw(t) is asymptotically stable.
We now give a numerical example to verify our theoretical results.
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730 I. ALI
Example 3.1. Let us consider a quaternionic matrix A =
\left[ 2 + \bfi \bfk \bfj
0 - \bfi \bfj
0 0 1 + \bfj
\right] . It is clear that
2 + \bfi , - \bfi and 1 + \bfj are three right eigenvalues of A. Thus the standard right eigenvalues of A are
2 + \bfi , \bfi and 1 + \bfi . Here, we obtain
3\sum
i=1
| \lambda i| = 1 +
\surd
2 +
\surd
5 \leq
3\sum
i=1
3\sum
j=1
| aij | = 4 +
\surd
2 +
\surd
5.
Furthermore, we also get
3\sum
i=1
| \lambda i| = 1 +
\surd
2 +
\surd
5 \leq 3 +
\surd
10 +
2
\surd
19 +
\surd
2
3
= \delta ,
where \delta =
\sum 3
i=1
\sum 3
j=1,j \not =i
| aij | +
\sum 3
i=1
\biggl( \bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
3
\bigm| \bigm| \bigm| \bigm| \biggr) + | \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)| . Thus, Theorem 3.1
is verified.
We next verify the results of the right spectral radius. The right spectral radius and the q-
determinant of A are
\rho r(A) = 2.2361 and | A| q = 10,
respectively. From (9), we have
3\sum
i=1
3\sum
j=1
| aij | - (2)
(| A| q)
1
4\Bigl( \sum 3
i=1
\sum 3
j=1
| aij |
\Bigr) 1
2
= 6.3644.
Thus, the inequality (9) is verified. From (10), we obtain
\xi 1 - (2)
(| A| q)
1
4
(\xi 1)
1
2
= 8.3881,
where \xi 1 =
\sum 3
i=1
\sum 3
j=1,j \not =i
| aij | +
\sum 3
i=1
\biggl( \bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
3
\bigm| \bigm| \bigm| \bigm| \biggr) + | \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)| . Hence, the inequality
(10) is verified. Since \| A| | 2F = 11, so, from (11), we get\Biggl[
\| A\| 2F - (2)
\biggl(
| A| q
\| A\| 2F
\biggr) 1
2
\Biggr] 1/2
= 3.0155.
Finally, the inequality (11) is verified.
4. Location of right eigenvalues of a quaternionic matrix. Firstly, in this section, we find a
minimal ball in 4D spaces which containing all the Geršgorin balls of a quaternionic matrix.
Theorem 4.1. Let A := (aij) \in Mn(\BbbH ). Then there must be a minimal ball in 4D spaces
containing all the Geršgorin balls of A:
\eta (A) =
\biggl\{
q \in \BbbH :
\bigm| \bigm| \bigm| \bigm| q - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl[
ri(A) +
\bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr] \biggr\} .
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BOUNDS FOR THE RIGHT SPECTRAL RADIUS OF QUATERNIONIC MATRICES 731
Proof. From Theorem 3.1, we have the following n Geršgorin balls for A:
Gi(A) = \{ q \in \BbbH : | q - aii| \leq ri(A)\} , 1 \leq i \leq n.
Based on the particle and centre gravity theorem, each Gi(A), 1 \leq i \leq n, can be treated as a particle
or a rigid body. Then the centre of all particles or rigid bodies is
1
n
\sum n
i=1
aii =
\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
. To find
the smallest ball, we set the following model:
\mathrm{m}\mathrm{i}\mathrm{n}
\bigm| \bigm| \bigm| \bigm| q - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| such that | qi - aii| \leq ri(A), 1 \leq i \leq n.
Since \bigm| \bigm| \bigm| \bigm| qi - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| qi - aii + aii -
\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \leq | qi - aii| +
\bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| ,\bigm| \bigm| \bigm| \bigm| qi - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \leq ri(A) +
\bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| ,
the solution to the above model is that\bigm| \bigm| \bigm| \bigm| q - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| = \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\bigm| \bigm| \bigm| \bigm| qi - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| ,\bigm| \bigm| \bigm| \bigm| q - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl[
ri(A) +
\bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr] =: R.
Thus, all the Geršgorin balls of A must belong to the smallest ball with radius R and center at
\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
. If we denote smallest ball by \eta (A), then
\eta (A) =
\biggl\{
q \in \BbbH :
\bigm| \bigm| \bigm| \bigm| q - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl[
ri(A) +
\bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr] \biggr\} .
Theorem 4.1 is proved.
Now, we turn to locate the right eigenvalues in the minimal ball \eta (A). In fact, a right eigenvalue
is not necessarily contained in a minimal ball \eta (A). For example, consider a quaternionic matrix
A =
\biggl[
\bfi 0
0 \bfj
\biggr]
. Here, - \bfi is a right eigenvalue of A but it is not contained in minimal ball \eta (A), that
is,
- \bfi /\in \eta (A) :=
\biggl\{
q \in \BbbH :
\bigm| \bigm| \bigm| \bigm| q - \bfi + \bfj
2
\bigm| \bigm| \bigm| \bigm| \leq 1
2
\surd
2
\biggr\}
.
Fortunately, we have the following theorem for right eigenvalues of a quaternionic matrix.
Theorem 4.2. Let A := (aij) \in Mn(\BbbH ). For every right eigenvalue \lambda of A there exists a
nonzero quaternion \alpha such that \alpha - 1\lambda \alpha (which is also a right eigenvalue) is contained in the
minimal ball \eta (A), that is,
\{ \alpha - 1\lambda \alpha : 0 \not = \alpha \in \BbbH \} \cap \eta (A) \not = \varnothing .
Proof. The proof follows from Theorems 3.1 and 4.1.
By definitions, Hermitian and \eta -Hermitian matrices have all the real diagonal entries. \eta -
Hermitian matrices arise widely in applications [7, 17, 18]. Thus, we present the following result.
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732 I. ALI
Theorem 4.3. Let A := (aij) \in Mn(\BbbH ) and aii \in \BbbR for all i. Then all the right eigenvalues of
A are contained in \eta (A).
Proof. Since all the right eigenvalues of a quaternionic matrix with all real diagonal entries are
contained in the union of n Geršgorin balls Gi(A), 1 \leq i \leq n. Therefore, the proof follows from
Theorem 4.1.
We now provide a numerical example to show the effectiveness of our result.
Example 4.1. Let us consider a quaternionic matrix A =
\left[ 1 \bfj \bfk
1 + \bfi 2 \bfi + \bfj
1 - \bfj \bfj + \bfk 4
\right] . Then the
complex adjoint matrix of A is given as
\Psi A =
\left[
1 0 0 0 1 \bfi
1 + \bfi 2 \bfi 0 0 1
1 0 4 - 1 1 + \bfi 0
0 - 1 \bfi 1 0 0
0 0 - 1 1 - \bfi 2 - \bfi
1 - 1 + \bfi 0 1 0 4
\right] .
The spectrum of \Psi A is \Lambda (\Psi A) = \{ 3.7627+1.0148\bfi , 2.6089+0.9307\bfi , 0.6283+0.6807\bfi , 3.7627 -
- 1.0148\bfi , 2.6089 - 0.9307\bfi , 0.6283 - 0.6807\bfi \} . Therefore, the right spectrum of A is \Lambda r(A) =
= [3.7627 + 1.0148\bfi ] \cup [2.6089 + 0.9307\bfi ] \cup [0.6283 + 0.6807\bfi ]. From Theorem 4.1, we obtain
\eta (A) =
\biggl\{
q \in \BbbH :
\bigm| \bigm| \bigm| \bigm| q - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl[
ri(A) +
\bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr] \biggr\} ,
\eta (A) =
\Biggl\{
q \in \BbbH :
\bigm| \bigm| \bigm| \bigm| q - 7
3
\bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\Biggl[
10
3
,
6
\surd
2 + 1
3
,
6
\surd
2 + 5
3
\Biggr] \Biggr\}
,
\eta (A) =
\biggl\{
q \in \BbbH :
\bigm| \bigm| \bigm| \bigm| q - 7
3
\bigm| \bigm| \bigm| \bigm| \leq 6
\surd
2 + 5
3
\biggr\}
.
By Theorem 4.3, we know that all the right eigenvalues of A should be contained in \eta (A). It is
clear that all the standard right eigenvalues \mu 1 = 3.7627 + 1.0148\bfi , \mu 2 = 2.6089 + 0.9307\bfi and
\mu 3 = 0.6283 + 0.6807\bfi are contained in \eta (A). Here, we can also easily see that\bigm| \bigm| \bigm| \bigm| \rho - 1\mu 1\rho -
7
3
\bigm| \bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| \mu 1 -
7
2
\bigm| \bigm| \bigm| \bigm| , \bigm| \bigm| \bigm| \bigm| \alpha - 1\mu 2\alpha - 7
3
\bigm| \bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| \mu 2 -
7
2
\bigm| \bigm| \bigm| \bigm| ,\bigm| \bigm| \bigm| \bigm| \beta - 1\mu 3\beta - 7
3
\bigm| \bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| \mu 3 -
7
2
\bigm| \bigm| \bigm| \bigm| \forall \rho , \alpha , \beta \in \BbbH \setminus \{ 0\} .
Hence all the right eigenvalues of A are contained in \eta (A). Thus, Theorem 4.3 is verified.
We are now ready to establish some results on a quaternionic matrix. In general, similar quater-
nionic matrices may have different traces follows from the following example.
Example 4.2. Let A =
\biggl[
\bfi \bfi
0 - \bfi
\biggr]
and B =
\left[ \bfi -
1
2
\bfk
1
2
\bfi
- 1
2
\bfi - \bfi - 1
2
\bfk
\right] . Then B = UHAU, where
U =
1\surd
2
\biggl[
1 - \bfi
- \bfj 1
\biggr]
is an unitary matrix but \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A) = 0 while \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(B) = - \bfk .
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BOUNDS FOR THE RIGHT SPECTRAL RADIUS OF QUATERNIONIC MATRICES 733
However, the following result is true.
Theorem 4.4. Let A := (aij) \in Mn(\BbbH ) be a central closed matrix. If A is similar to quater-
nionic matrix B. Then B is also a central closed matrix. Moreover, \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A) = \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(B).
Proof. Since A is similar to B, then there exists a nonsingular quaternion matrix P such that
A = PBP - 1. Also A is central closed matrix, then there exists a nonsingular quaternionic matrix
Q such that A = QDQ - 1, where D = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\lambda 1, \lambda 2, . . . , \lambda n) with the real standard right eigenvalues
\lambda i. From the above, we have
QDQ - 1 = PBP - 1 \Rightarrow B = P - 1DQ - 1P.
Setting P - 1Q = T, then we obtain B = TDT - 1. Hence, B is also a central closed matrix.
For second part: From Lemma 2.1, we have \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A) =
\sum n
i=1
\lambda i = \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(D). Moreover,
\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(B) =
\sum n
i=1
\lambda i = \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(D). It follows from the above that \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A) = \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(B).
Theorem 4.5. Let A := (aij) \in Mn(\BbbH ) be central closed matrix. If B1, B2, . . . , Bs are similar
to A, then we can derive a minimal ball in 4D spaces which contain all Geršgorin balls of at least
one matrix among B1, B2, . . . , Bs and A. That is,
Gmin(A) =
\biggl\{
q \in \BbbH :
\bigm| \bigm| \bigm| \bigm| q - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{i}\mathrm{n}
1\leq k\leq s
\biggl\{
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl[
ri(A) +
\bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr] ,
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl[
ri(Bk) +
\bigm| \bigm| \bigm| \bigm| bii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr] \biggr\} .
Proof. Since B1, B2, . . . , Bs are similar to A, then from Theorem 4.4, we have \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(Bk) =
= \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A), 1 \leq k \leq s. It reveals that the balls \eta (B1), \eta (B2), . . . , \eta (Bs) and \eta (A) are concentric
balls whose centers at
\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
. Therefore, it only needs us to find the ball with the smallest radius
from s+ 1 concentric balls.
If we denote Gmin(A) a ball with the smallest radius, then we have
Gmin(A) =
\biggl\{
q \in \BbbH :
\bigm| \bigm| \bigm| \bigm| q - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{i}\mathrm{n}
1\leq k\leq s
\biggl\{
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl[
ri(A) +
\bigm| \bigm| \bigm| \bigm| aii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr] ,
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl[
ri(Bk) +
\bigm| \bigm| \bigm| \bigm| bii - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)
n
\bigm| \bigm| \bigm| \bigm| \biggr] \biggr\} .
From the above it is clear that the radius and center of the smallest ball can be determined by entries
of B1, B2, . . . , Bs and A.
Theorem 4.6. Let A := (aij) \in Mn(\BbbH ) be central closed matrix and \lambda 1, \lambda 2, . . . , \lambda n be n right
eigenvalues of A. If B1, B2, . . . , Bs are similar to A, then \lambda 1, \lambda 2, . . . , \lambda n are contained in Gmin(A).
Proof. Since central closed quaternionic matrices have all real right eigenvalues, then, from
Theorem 3.1, all the right eigenvalues of A are contained in
\bigcup n
i=1
Gi(A). Therefore, from Theorem
4.5, we have the required result.
Example 4.3. Consider a central closed quaternionic matrix A =
\left[
1 - \bfi - \bfj \bfk
\bfi 1 - 2\bfk \bfj
\bfj 2\bfk 7 - \bfi
- \bfk - \bfj \bfi 1
\right] .
Then the complex adjoint matrix of A is given as
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734 I. ALI
\Psi A =
\left[
1 - \bfi 0 0 0 0 - 1 \bfi
\bfi 1 0 0 0 1 - 2\bfi 1
0 0 7 - \bfi 1 2\bfi 0 0
0 0 \bfi 1 - \bfi - 1 0 0
0 0 1 \bfi 1 \bfi 0 0
0 - 1 - 2\bfi - 1 - \bfi 1 0 0
- 1 2\bfi 0 0 0 0 7 \bfi
- \bfi 1 0 0 0 0 - \bfi 1
\right]
.
The spectrum of \Psi A is \Lambda (\Psi A) = \{ - 1, 1, 2, 8\} . Consequently, the right spectrum of A is \Lambda r(A) =
= \{ - 1, 1, 2, 8\} . Since A is a central closed matrix, so A is similar to the diagonal matrix D =
= \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}( - 1, 1, 2, 8). From Theorem 4.6, we get
Gmin(A) =
\biggl\{
q \in \BbbH :
\bigm| \bigm| \bigm| \bigm| q - 5
2
\bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{i}\mathrm{n}
\biggl[
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl\{
9
2
,
11
2
, 9
\biggr\}
, \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl\{
7
2
,
3
2
,
1
2
,
11
2
\biggr\} \biggr] \biggr\}
,
Gmin(A) =
\biggl\{
q \in \BbbH :
\bigm| \bigm| \bigm| \bigm| q - 5
2
\bigm| \bigm| \bigm| \bigm| \leq 11
2
\biggr\}
.
Here, all the right eigenvalues of A are contained in Gmin(A). Hence, Theorem 4.6 is verified.
Finally, we present a numerical example which shows that our inclusion region Gmin(A) (defined
in Theorem 2.1) is potentially sharper than the inclusion region G(A) (defined in Theorem 4.5) for
some quaternionic matrices.
Example 4.4. Let us consider a central closed quaternionic matrix
A =
\left[ 1 3 + 9\bfi - 12\bfj + 10\bfk 13\bfi - 10\bfj - 7\bfk
3 - 9\bfi + 12\bfj - 10\bfk 3 5\bfi - 7\bfj + 6\bfk
- 13\bfi + 10\bfj + 7\bfk - 5\bfi + 7\bfj - 6\bfk 2
\right] .
Then, the complex adjoint matrix of A is given as
\Psi A =
\left[
1 3 + 9\bfi 13\bfi 0 - 12 + 10\bfi - 10 - 7\bfi
3 - 9\bfi 3 5\bfi 12 - 10\bfi 0 - 7 + 6\bfi
- 13\bfi - 5 2 10 + 7\bfi 7 - 6 0
0 12 + 10\bfi 10 - 7\bfi 1 3 - 9\bfi - 13\bfi
- 12 - 10\bfi 0 7 + 6\bfi 3 + 9\bfi 3 - 5\bfi
- 10 + 7\bfi - 7 - 6\bfi 0 13\bfi 5\bfi 2
\right] .
The right spectrum of A is \Lambda r(A) = \{ - 25.3430, 1.4493, 29.8937\} . We can easily see that A is sim-
ilar to D := \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}( - 25.3430, 1.4493, 29.8937). From Theorems 2.1 and 4.6, we have the following
balls:
G(A) = \{ z \in \BbbH : | z - 2| \leq 31.8957\} and Gmin(A) = \{ z \in \BbbH : | z - 2| \leq 27.8937\} .
From the above balls, it is clear that Gmin(A) \subset G(A). Thus our estimation is could be sharp for
some quaternionic matrices.
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BOUNDS FOR THE RIGHT SPECTRAL RADIUS OF QUATERNIONIC MATRICES 735
References
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Received 25.04.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
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| id | umjimathkievua-article-6018 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:19Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a1/97671d057fde74d1c091fad13ee966a1.pdf |
| spelling | umjimathkievua-article-60182022-03-26T11:01:50Z Bounds for the right spectral radius of quaternionic matrices Bounds for the right spectral radius of quaternionic matrices Ali, I. Ali, I. UDC 517.5&nbsp; In this paper&nbsp; we present bounds for the sum of the moduli of right eigenvalues of a quaternionic matrix. As a consequence, we obtain bounds for the right spectral radius of a quaternionic matrix. We also present a minimal ball in 4D spaces which contains all the Gersgorin balls of a quaternionic matrix. As an application, we introduce the estimation for the right ˇ eigenvalues of quaternionic matrices in the minimal ball. Finally, we suggest some numerical examples to illustrate of our results. УДК 517.5&nbsp; Знайдено граничнi оцiнки для сум модулiв правих власних значень кватернiонних матриць. Як наслiдок, отримано оцiнки для правого спектрального радiуса таких матриць. У чотиривимiрних просторах знайдено мiнiмальний шар, який мiстить всi шари Гершгорiна матрицi кватернiонiв. Як застосування, запропоновано оцiнку для правих власних значень матриць кватернiонiв. Також наведено приклади для iлюстрацiї цих результатiв. Institute of Mathematics, NAS of Ukraine 2020-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6018 10.37863/umzh.v72i6.6018 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 6 (2020); 723-735 Український математичний журнал; Том 72 № 6 (2020); 723-735 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6018/8710 |
| spellingShingle | Ali, I. Ali, I. Bounds for the right spectral radius of quaternionic matrices |
| title | Bounds for the right spectral radius of quaternionic matrices |
| title_alt | Bounds for the right spectral radius of quaternionic matrices |
| title_full | Bounds for the right spectral radius of quaternionic matrices |
| title_fullStr | Bounds for the right spectral radius of quaternionic matrices |
| title_full_unstemmed | Bounds for the right spectral radius of quaternionic matrices |
| title_short | Bounds for the right spectral radius of quaternionic matrices |
| title_sort | bounds for the right spectral radius of quaternionic matrices |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6018 |
| work_keys_str_mv | AT alii boundsfortherightspectralradiusofquaternionicmatrices AT alii boundsfortherightspectralradiusofquaternionicmatrices |