Research Article
The General Solution of Quaternion Matrix Equation Having
-Skew-Hermicity and Its Cramer’s Rule
Abdur Rehman,
1
Ivan Kyrchei ,
2
Ilyas Ali,
1
Muhammad Akram,
1
and Abdul Shakoor
3
1
University of Engineering & Technology, Lahore, Pakistan
2
Pidstrygach Institute for Applied Problems of Mechanics and Mathematics of NASU, Lviv, Ukraine
3
Khawaja Fareed University of Engineering & Information Technology, Rahim Yar Khan, Pakistan
Correspondence should be addressed to Ivan Kyrchei; st260664@gmail.com
Received 18 March 2019; Accepted 10 June 2019; Published 30 July 2019
Academic Editor: Rafaele Solimene
Copyright © 2019 Abdur Rehman et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We determine some necessary and sufcient conditions for the existence of the -skew-Hermitian solution to the following system
− ()
∗
+
∗
+
∗
= , = −
∗
, = −
∗
over the quaternion skew feld and provide an explicit expression
of its general solution. Within the framework of the theory of quaternion row-column noncommutative determinants, we derive
its explicit determinantal representation formulas that are an analog of Cramer’s rule. A numerical example is also provided to
establish the main result.
1. Introduction
Troughout this paper, the real number feld and the complex
feld are denoted by R and C, respectively, and H for the
quaternion algebra
H = {ℎ
0
+ℎ
1
i +ℎ
2
j +ℎ
3
k | i
2
= j
2
= k
2
= ijk
= −1, ℎ
0
,ℎ
1
,ℎ
2
,ℎ
3
∈ R}.
(1)
Te set of all matrices of dimension × over H is
represented by H
×
. An identity matrix with conformable
size is denoted by . For any matrix over H, R() and
N() stand for the column right space and the row lef space
of , respectively. [R()] denotes the dimension of R().
By [1], we have [R()] = [N()], which is known as
rank of and denoted by (). For any quaternion ℎ=
ℎ
0
+ℎ
1
i +ℎ
2
j +ℎ
3
k, its conjugate is ℎ=ℎ
0
−ℎ
1
i −ℎ
2
j −ℎ
3
k.
So,
∗
represents the conjugate transpose of .
†
means the
Moore-Penrose inverse of ∈ H
×
, i.e., the exclusive matrix
∈ H
×
satisfying
= ,
= ,
()
∗
= ,
()
∗
= .
(2)
More results on generalized inverses can be seen in [2, 3].
Furthermore,
=−
†
and
=−
†
are couple
of the projectors induced by , respectively. It is evident that
=
∗
=
†
=
2
and
=
∗
=
†
=
2
.
Te notion of quaternions was frst explored by an
Irish mathematician Sir William Rowan Hamilton in [4].
Quaternions have ample use in diverse areas of mathematics
like computation, geometry and algebra; see, e.g. [5–8]. Te
researcher in [9] used quaternion in the feld of computer
graphics. Very recently, quaternion matrices have secured
a vital role in control theory, mechanics, altitude control,
quantum physics and signal processing; see, e.g., [10, 11]. In
skeletal animation systems, quaternions are mostly practiced
to interpolate between joint orientations specifed with key
frames or animation curves [12]. Te comprehensive study on
quaternions can be found in [13].
Hindawi
Mathematical Problems in Engineering
Volume 2019, Article ID 7939238, 25 pages
https://doi.org/10.1155/2019/7939238