On the solutions of the quaternion interval systems [x]=[A] [x]+ [b]

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On the solutions of the quaternion interval systems

½x ¼ ½A½x þ ½b

Cennet Bolat

a,⇑

, Ahmet _Ipek

b

a

Mustafa Kemal University, Faculty of Art and Science, Department of Mathematics, Tayfur Sökmen Campus, Hatay, Turkey

b

Karamanog˘lu Mehmetbey University, Faculty of Kamil Özdag˘ Science, Department of Mathematics, 70100 Karaman, Turkey

a r t i c l e

i n f o

Keywords: Intervals Quaternions

The systems of equations

a b s t r a c t

It is known that linear matrix equations have been one of the main topics in matrix theory and its applications. The primary work in the investigation of a matrix equation (system) is to give solvability conditions and general solutions to the equation(s). In the present paper, for the quaternion interval system of the equations deﬁned by ½x ¼ ½A½x þ ½b, where ½A is a quaternion interval matrix and ½b and ½x are quaternion interval vectors, we derive a necessary and sufﬁcient criterion for the existence of solutions ½x. Thus, we reduce the existence of a solution of this system in quaternion interval arithmetic to the existence of a solution of a system in real interval arithmetic.

1. Introduction

Interval arithmetic was ﬁrst suggested by Dwyer in 1951[9]. Development of interval arithmetic as a formal system and evidence of its value as a computational device was provided by Moore[19,20]. After this motivation and inspiration, several authors such as Alefeld and Herzberger[1], Hansen[12–14]and Neumaier[21], etc. have studied interval arithmetic.

Many practical problems ﬁnally lead to systems of the linear equations

Cx ¼ b; C 2 Rnn_; _{b 2 R}n_: _ð1Þ

Mostly C is regular and therefore the system (1) is uniquely solvable. When solving linear systems of equations1the Richardson splitting C ¼ I A (see the discussion in Sections (3.3) and (3.4) of[23]) leads to the equivalent ﬁxed point form

x ¼ Ax þ b ½2:

Recently, a large number of papers have studied interval matrices and interval systems [[3–21]]. By Arndt and Mayer’s paper[5], the question on the existence of solutions of the system

½x ¼ ½A½x þ ½b; ð2Þ

where ½A is a real interval matrix and ½b and ½x are real interval vectors, is completely clariﬁed. Furthermore the question on the convergence of powers of ½A is answered for real interval matrices except two minor cases (cf. [3,4,17]). In[6], for complex interval matrices ½A and complex interval vectors ½b and ½x, a necessary and sufﬁcient criterion is given for the existence of a solution of(2)for which j½Aj is reducible with

q

ðj½AjÞ P 1.

Quaternions were introduced in the mid-nineteenth century by Hamilton[10,11]as an extension of complex numbers and as a tool for manipulating 3-dimensional vectors. Indeed Maxwell used them to introduce vectors in his exposition of

⇑Corresponding author.

E-mail addresses:bolatcennet@gmail.com,cbolat@mku.edu.tr(C. Bolat),dr.ahmetipek@gmail.com(A. _Ipek).

Contents lists available atScienceDirect

Applied Mathematics and Computation

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electromagnetic theory[18]. Many authors over the past 20 years have ‘‘rediscovered’’ the application of quaternions. These papers may be supplemented with a wealth of on-line resources[22,24].

In 2010 Bolat and _Ipek[7]extend the powerful ideas in that study to the quaternions with real interval coefficients. Many number of concepts and techniques that we learned in a standard setting for quaternions with real number coefficients, real intervals and matrices can be carried over to quaternion interval numbers. In that paper, firstly it is defined the quaternion intervals set and the quaternion interval numbers, secondly they present the representation vector and matrix for quater-nion interval numbers, and then investigate some algebraic properties of these representations, which the representation matrix is called the fundamental matrix, and finally they compute the determinant, norm, inverse, trace, eigenvalues and eigenvectors of the representation matrix established for a general quaternion interval number.

In 2011 Bolat and _Ipek[8]derive a necessary and sufﬁcient criterion for the convergence of powers of quaternion interval matrices ½A to a limit which may differ from O. Generalizing former results we allow now the absolute value j½Aj of ½A to be reducible with minor additional restrictions.

In the present paper, for the quaternion interval system of the equations defined by(2), where ½A is a quaternion interval matrix and ½b and ½x are quaternion interval vectors, we derive a necessary and sufficient criterion for the existence of solutions ½x. Thus, we reduce the existence of a solution of this system in quaternion interval arithmetic to the existence of a solution of a system in real interval arithmetic. In addition, we give a necessary and sufficient criterion for the conver-gence of power of ½A.

The remainder of this paper is organized as follows. In Section2 notation and preliminary results are presented. In Section2.1, basic quaternion algebra is introduced. Some notation and basic facts are listed for real, complex and quaternion intervals being used in the sequel in Section2.2. In Section3our new results are given.

2. Some preliminaries

In this section, we introduce some deﬁnitions, notations and basic properties which we need to use in the presentations and proofs of our main results in Section3.

2.1. Quaternion numbers

In this subsection, we introduce the deﬁnitions of the quaternion and quaternion matrix and their basic properties that will be used in the sequel. Basic quaternion algebra is well covered in Hamilton’s papers[10,11], which are both accessible and readable.

The original notation for quaternions[10]paralleled the convention for complex numbers

q ¼ q0u þ q1i þ q2j þ q3k;

which obey the conventional algebraic rule for addition and multiplication by scalars (real numbers) and which obey an associative non-commutative rule for multiplication where u is the identity element and

i2¼ j2¼ k2¼ u; ij ¼ ji ¼ k; jk ¼ kj ¼ i; ki ¼ ik ¼ j: ð3Þ

It is frequently useful to regard quaternions as an ordered set of 4 real quantities which we write as

q ¼ q½ 0;q1;q2;q3 ð4Þ

or as a combination of a scalar and a vector

q ¼ q½ 0;q; ð5Þ

where q ¼ q½ ₁;q₂;q₃. A ‘‘scalar’’ quaternion has zero vector part and we shall write this as q½ 0;0 ¼ q0u ¼ 0. A ‘‘pure’’

quater-nion has zero scalar part 0; q½ . In the scalar–vector representation, multiplication becomes

pq ¼ pð 0q0 p q; p0q þ q0p þ p qÞ;

where ‘‘’’ and ‘‘’’ are the vector dot and cross products. The conjugate of a quaternion is given by

q ¼ q½ 0;q;

the squared norm of a quaternion is

q j j2¼ qq ¼ q2 0þ q 2 1þ q 2 2þ q 2 3 and its inverse is

q1_¼ q

q j j2:

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An alternative way[15]to deﬁne quaternions is to consider the subset of the ring M2ð Þ of 2 2 matrices with complexC numbers entries: Q0¼ a1 a2 a2 a1 :a1¼ q0þ q1i; a2¼ q2þ q3i 2 C ð6Þ or as a 4 4 real matrix, Q0¼ q0 q1 q2 q3 q1 q0 q3 q2 q2 q3 q0 q1 q3 q2 q1 q0 0 B B B @ 1 C C C A:qi2 R; i ¼ 0; 1; 2; 3 8 > > > < > > > : 9 > > > = > > > ; : ð7Þ

Q0is a subring of M2ð Þ under the operations of MC 2ð Þ.C

Remark 1. Q and Q0_{are essentially the same. In fact, let}

w :q ¼ a1þ a2j 2 Q ! q0¼

a1 a2

a2 a1

2 Q0_:

wis bijective and preserves the operations. Furthermore, qj j2¼ det q0_{, and the eigenvalues of q}0_{are Req Imq}_j _ji.

2.2. Interval numbers and matrices

In this stage, we list some notation and basic facts for real, complex and quaternion intervals being used in the sequel. Interval arithmetics can be deﬁned in different ways. We use a general approach where different arithmetics can be deﬁned without much effort.

Deﬁnition 2. A real interval is the set

a

½ ¼ a; a½ ¼ x 2 R : a 6 x 6 af g:

We denote the set of all real intervals by I Rð Þ. For two intervals a½ ; b½ 2 I Rð Þ the absolute value, the radius, the Hausdorff distance and the operations are deﬁned by

a ½ j j ¼ max a; af g ¼ max c2 a½ j j;c rad að½ Þ ¼ a að Þ=2; q að½ ; b½ Þ ¼ maxnja bj; a b o; a ½ þ b½ ¼ a þ b; a þ bh i; a ½ b½ ¼ a b; a bh i; a

½ b½ ¼ min ab; ab; ab; abh ;max ab; ab; ab; ab i:

If a½ 2 I Rð Þ contains only one element a then, trivially, a ¼ a ¼ a. In this case we identify a½ with its element writing a

½ ¼ a and calling a½ degenerate or point interval. Analogously, we deﬁne degenerate interval vectors and degenerate inter-val matrices as point matrices. For notation and further details about real interinter-val numbers and their arithmetics we refer to

[1,5].

Deﬁnition 3 (Complex interval number). A complex interval is the set

a

½ ¼ a½ re þ i a½ im ¼ x 2 C : R xf ð Þ 2 a½ re 2 I Rð Þ; I xð Þ 2 a½ im 2 I Rð Þg:

We denote the set of all complex intervals by I Cð Þ. For two intervals a½ ; b½ 2 I Cð Þ the absolute value, the radius, the Haus-dorff distance and the operations are deﬁned by

a ½

j j ¼ aj½ rej þ aj½ imj ¼ max

c2 a½ ðjI cð Þj þ R cj ð ÞjÞ;

rad að½ Þ ¼ rad að½ reÞ þ rad að½ imÞ;

q að½ ; b½ Þ ¼ q að½ re; b½ reÞ þ q að½ im; b½ imÞ; a ½ þ b½ ¼ að½ re þ b½ reÞ þ i að½ im þ b½ imÞ; a ½ b½ ¼ að½ re b½ reÞ þ i að½ im b½ imÞ; a ½ b½ ¼ að½ re b½ re a½ im b½ imÞ þ i að½ im b½ re þ a½ re b½ imÞ; where we use the standard real interval arithmetic[6].

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Deﬁnition 4 (Quaternion interval number). A quaternion interval is the set

q

½ ¼ q½ þ q0 ½ i þ q1 ½ j þ q2 ½ k ¼ a3 f 0þ a1i þ a2j þ a3k : as2 q½ 2 I Rs ð Þ; s ¼ 0; 1; 2; 3g: where i; j and k elements satisfy(3)condition[7].

We denote the set of all quaternion intervals by I Hð Þ. For two quaternion intervals

a

½ ¼ a½ þ a0 ½ i þ a1 ½ j þ a2 ½ k 2 I H3 ð Þ;

b

½ ¼ b½ þ b0 ½ i þ b1 ½ j þ b2 ½ k 2 I H3 ð Þ;

the absolute value, the radius, the Hausdorff distance and the operations are deﬁned by

a ½ j j ¼ aj½ 0j þ aj½ 1j þ aj½ 2j þ aj½ 3j ¼ max c¼c0þc1iþc2jþc3k2 a½ c0 j j þ cj j þ c1 j j þ c2 j j3 f g;

rad að½ Þ ¼ radð a½ Þ þ radð a0 ½ Þ þ radð a1 ½ Þ þ radð a2 ½ Þ;3

q að½ ; b½ Þ ¼ q að½ ; b0 ½ 0Þ þ q að½ ; b1 ½ 1Þ þ q að½ ; b2 ½ 2Þ þ q að½ ; b3 ½ 3Þ; a ½ b½ ¼ ð a½ b0 ½ Þ þ a0 ð½ b1 ½ 1Þi þ að½ b2 ½ 2Þj þ að½ b3 ½ 3Þk; a ½ b½ ¼ að 0b0 a1b1 a2b2 a3b3Þ þ að 1b0þ a0b1 a3b2þ a2b3Þi þ að 2b0þ a3b1þ a0b2 a1b3Þj þ að 3b0 a2b1þ a1b2þ a0b3Þk:

In addition to the notations IðRÞ and IðCÞ, for the real and complex interval matrices we respectively use the following notations:

½A ¼ ½A; A ¼ ð½aijÞ_i;j¼1;...;n2 IðRnnÞ for ½aij2 IðRÞ; i; j ¼ 1; . . . ; n; and

½A ¼ ½A0 þ i½A1 2 IðCnnÞ for ½A0; ½A1 2 IðRnnÞ:

We assume some familiarity when working with these deﬁnitions. For details see, e.g., the introductory chapters of[5,6]. In addition to the notations IðHÞ, for the quaternion interval matrix we use the following notation:

½A ¼ ½A0 þ ½A1i þ ½A2j þ ½A3k 2 IðHnnÞ for ½As 2 IðRnnÞ; s ¼ 0; 1; 2; 3:

If the quaternion interval matrix ½A is partitioned in blocks ½A_ij, i; j ¼ 1; . . . ; s, we also write ½A ¼ ð½A_ijÞ_i;j¼1;...;s. For quaternion interval vectors and matrices the absolute value, the radius, and the distance are deﬁned entrywise, for instance

½A j j ¼ ½aij 2 Rnn_.

At this point we want to stress that by deﬁnition the absolute value of a degenerated quaternion interval, i.e. an interval with radð½aÞ ¼ 0, is not the Euclidian norm of the quaternion number. We must pay attention to this fact when we calculate j½Aj in quaternion interval arithmetic for an ½A which contains degenerate intervals. By convention the appropriate deﬁni-tion of the absolute value is used in our formulas. So we have to apply

a0þ a1i þ a2j þ a3k

j j ¼ aj j þ a0 j j þ a1 j j þ a2 j j3 to degenerate intervals in quaternion interval arithmetic. 3. New results

In this section we present some new results. We start with the deﬁnition of the representation matrix of a quaternion interval matrix.

Deﬁnition 5. Let ½A ¼ ½A0 þ ½A1i þ ½A2j þ ½A3k 2 IðHmnÞ. By (6) and (7), the matrix representation of the quaternion interval matrix ½A is the matrix

Rð½AÞ ¼ ½A0 þ ½A1i ½A2 þ ½A3i ½A2 þ ½A3i ½A0 ½A1i

¼ A0 ½ ½A1 ½A2 ½A3 A½ 1 ½A0 A½ 3 ½A2 A½ 2 ½A3 ½A0 A½ 1 A½ 3 A½ 2 ½A1 ½A0 0 B B B @ 1 C C C A½7:

Lemma 6. For three quaternion interval matrices ½A; ½B 2 IðHmn

Þ and ½C 2 IðHnpÞ, it is

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and

Rð½A ½CÞ ¼ Rð½AÞ Rð½CÞ:

Proof. For ½A; B½ 2 IðHmn_{Þ the addition and subtraction of the interval matrices Rð½AÞ and Rð½BÞ is written as} Rð½AÞ Rð½BÞ ¼ ½A0 þ ½A1i ½A2 þ ½A3i

½A2 þ ½A3i ½A0 ½A1i

½B0 þ ½B1i ½B2 þ ½B3i ½B2 þ ½B3i ½B0 ½B1i

¼ ½A0 ½B0 þ ½Að 1 ½B1Þi ð½A2 ½B2Þ þ ½Að 3 ½B3Þi ½Að 2 ½B2Þ þ ½Að 3 ½B3Þi ½A0 ½B0 ½Að 1 ½B1Þi

¼ Rð½A ½BÞ:

Similarly, by matrix multiplication, for multiplication of the interval matrices Rð½AÞ and Rð½CÞ we have

Rð½AÞ Rð½CÞ ¼ Rð½A ½CÞ:

The following theorem reduces the existence of a solution of the system

½x ¼ ½A½x þ ½b ð8Þ

with ½A ¼ ½A0 þ ½A1i þ ½A2j þ ½A3k 2 IðHnnÞ; ½b ¼ ½b0 þ ½b1i þ ½b2j þ ½b3k 2 IðHnÞ and ½x 2 IðHnÞ, in quaternion interval

arithmetic to the existence of a solution of the system

½x ¼ R Að½ Þ½x þ ½b0 ½b1 ½b2 ½b3 2 6 6 6 4 3 7 7 7 5 ð9Þ

in real interval arithmetic. Also, the following theorem establishes the link between the solution of real system(9)by the solution of quaternion interval system(8).

Theorem 7. The quaternion interval system(8)has a solution if and only if real system(9)has a solution. For every solution x

½ ¼ xð½ 0;½ x1;½ x2;½ x3Þ T

2 IðH4nÞ of9)½ y¼ x½ 0þ x½ 1i þ x½ 2j þ x½ 3k 2 IðHnÞ is a solution(8)and conversely. Proof. ‘‘)’’ : y½ _{¼ x} 0 ½ _{þ x} 1 ½ _{i þ x} 2 ½ _{j þ x} 3

½ _{k 2 IðH}n_{Þ be a solution of}₍₈₎_{. Then we get} y

½ ¼ A½ y½ þ b½

and from where we write

R yð½ Þ ¼ R Að½ y½ þ b½ Þ:

ByLemma 6, we have

R yð½ Þ ¼ R Að½ Þ R yð½ Þ þ R bð½ Þ

or

R yð½ Þ ¼ R Að½ Þ ðR yð½ ÞÞ R bð½ Þ: ð10Þ

From Eq.(10), we reach

x0 ½ x1 ½ x2 ½ x3 ½ 2 6 6 6 6 6 4 3 7 7 7 7 7 5 ¼ R Að½ Þ x0 ½ x1 ½ x2 ½ x3 ½ 2 6 6 6 6 6 4 3 7 7 7 7 7 5 þ b½ 0 b1 ½ b2 ½ b3 ½ 2 6 6 6 6 6 4 3 7 7 7 7 7 5 : ð11Þ For x½ _{¼ x} 0 ½ _;_x 1 ½ _;_x 2 ½ _;_x 3 ½

ð ÞTwe obtain from Eq.(11)

x ½ ¼ R Að½ Þ x½ þ ½b0 ½b1 ½b2 ½b3 2 6 6 6 6 4 3 7 7 7 7 5:

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This equation is the assertion.

‘‘(’’ : Now, let u½ ¼ uð½ 0;½u1;½u2;½u3Þ T

2 IðH4nÞ be a solution of the system(9). Then we write

½u0 ½u1 ½u2 ½u3 2 6 6 6 4 3 7 7 7 5¼ R Að½ Þ ½u0 ½u1 ½u2 ½u3 2 6 6 6 4 3 7 7 7 5þ ½b0 ½b1 ½b2 ½b3 2 6 6 6 4 3 7 7 7 5: ð12Þ

From Eq.(12)we reach the following equalities

½u0¼ ½A0½u0þ ½A1½u1þ ½A2½u2þ ½A3½u3 ½b0;

½u1¼ ½A1½u0þ ½A0½u1 ½A3½u2þ ½A2½u3þ ½b1;

½u2¼ ½A2½u0þ ½A3½u1þ ½A0½u2 ½A1½u3þ ½b2;

½u3¼ ½A3½u0 ½A2½u1þ ½A1½u2þ ½A0½u3þ ½b3: From these equalities it is written the system

½u0 ½u1 ½u2 ½u3

½u1 ½u0 ½u3 ½u2

½u2 ½u3 ½u0 ½u1

½u3 ½u2 ½u1 ½u0

2 6 6 6 6 4 3 7 7 7 7 5¼

½A0 ½A1 ½A2 ½A3

½A1 ½A0 ½A3 ½A2

½A2 ½A3 ½A0 ½A1

½A3 ½A2 ½A1 ½A0

2 6 6 6 6 4 3 7 7 7 7 5

½u0 ½u1 ½u2 ½u3

½u1 ½u0 ½u3 ½u2

½u2 ½u3 ½u0 ½u1

½u3 ½u2 ½u1 ½u0

2 6 6 6 6 4 3 7 7 7 7 5 þ ½b0 ½b1 ½b2 ½b3 ½b1 ½b0 ½b3 ½b2 ½b2 ½b3 ½b0 ½b1 ½b3 ½b2 ½b1 ½b0 2 6 6 6 6 4 3 7 7 7 7 5: ð13Þ

It is obviously seen that from Eq.(13)we have for z½ ¼ u½ 0þ u½ 1i þ u½ 2j þ u½ 3k R zð½ Þ ¼ R Að½ Þ ðR zð½ ÞÞ R bð½ Þ

or

R zð½ Þ ¼ R Að½ Þ R zð½ Þ þ R bð½ Þ: ð14Þ

ByLemma 6and Eq.(14)we can write

R zð½ Þ ¼ Rð A½ z½ þ b½ Þ:

Therefore byRemark 1we obtain the result

z

½ ¼ A½ z½ þ b½ ;

which completes the proof. h

We will illustrate our result through one numerical example and all computations are carried out by Maple in this example.

Example 8. For degenerate quaternion interval matrix A½ ¼ 1 i

j 1

2 H I R 22 and degenerate quaternion interval vectors b½ ¼ 1 1 ;½ ¼y ½y1 y2 ½

2 H I R 2, the degenerate quaternion interval system

y

½ ¼ A½ y½ þ b½ ð15Þ

has a solution such that

y ½ ¼ j

i

with quaternion algebra. That is, for y½ 0 ¼ 0 0 ; ½ y1 ¼ 0 1 ; ½ y2 ¼ 1 0 and y½ 3 ¼ 0 0 , y ½ ¼ y½ 0 þ y½ 1 _{i þ y} 2 ½ j þ y½ 3 _k: Now we let consider the real interval system

½z ¼ R Að½ Þ½z þ 1 1 0 0 0 0 0 0ð ÞT

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z ½ ¼ 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 B B B B B B B B B B B B B @ 1 C C C C C C C C C C C C C A z ½ þ 1 1 0 0 0 0 0 0 0 B B B B B B B B B B B B B @ 1 C C C C C C C C C C C C C A :

If this system is solved in real arithmetic, the solution of the system is obtained as

z ½ _¼ y½ 0 y1 ½ y2 ½ y3 ½ 0 B B B @ 1 C C C A: 4. Conclusions

It is known that linear matrix equations have been one of the main topics in matrix theory and its applications. The pri-mary work in the investigation of a matrix equation (system) is to give solvability conditions and general solutions to the equation(s). We have derived some new necessary and sufﬁcient conditions for the existence and the expression of the gen-eral solution to system(8).

Acknowledgement

The authors would like to thank the editor and the reviewers for their valuable comments and helpful suggestions, which improved the paper.

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