Cramer’s rules for various solutions to some restricted quaternionic linear systems

J. Appl. Math. Comput.DOI 10.1007/s12190-014-0793-2

ORIGINAL RESEARCH

Cramer’s rules for various solutions to some restrictedquaternionic linear systems

Guang-Jing Song · Haixia Chang ·Zhongcheng Wu

Received: 18 March 2014© Korean Society for Computational and Applied Mathematics 2014

Abstract In this paper, we show some new necessary and sufficient conditions for theexistences of the generalized inverses A(2)

rT1,S1and A(2)

lT2,S2over the quaternion skew field

by checking the nonsingularity of some matrices instead of computing the direct sumof some quaternionic vector spaces. We also derive a series of concise determinantalrepresentations of these generalized inverses. In addition, we give some condensedCramer’s rules for the general solutions, the least squares solutions and the approximatesolutions to some restricted quaternionic systems of linear equations, respectively.

Keywords Quaternion matrix · Cramer’s rule · Generalized inverse A(2)T ,S ·

Determinant · System of linear equations

Mathematical Subject Classification 15A09 · 15A24

1 Introduction

Throughout, we denote the real number field by R, the set of all m × n matrices overthe quaternion algebra

G.-J. Song (B)School of Mathematics and Information Sciences, Weifang University,Weifang 261061, People’s Republic Chinae-mail: [email protected]

H. ChangDepartment of Applied Mathematics, Shanghai Finance University,Shanghai 201209, People’s Republic China

Z. WuSchool of fundamental studies, Shanghai University of Engineering Science,Shanghai 201620, People’s Republic China

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G.-J. Song et al.

H = {a0 + a1i + a2 j + a3k | i2 = j2 = k2 = i jk = −1, a0, a1, a2, a3 ∈ R}

by Hm×n, the identity matrix with the appropriate size by I. For A ∈ H

m×n, thesymbols A∗ stands for the conjugate transpose of A. The Moore–Penrose inverse ofA, denoted by A†, is the unique matrix X ∈ H

n×m satisfying the Penrose equations

(1) AX A = A, (2) X AX = X, (3) (AX)∗ = AX, (4) (X A)∗ = X A.

In the study of theory and numerical computations of quaternionic quantum theory,in order to well understand the perturbation theory [1], experimental proposals [2] andtheoretical discussions [3] underlying the quaternionic formulations of the Schrodingerequation and so on, one often meets problems of approximate solutions of quaternionproblems, such as the approximate solution of the quaternion linear equations

Ax = b, (1.1)

that is appropriate when there is error in the b, i.e. quaternionic least square (QLS)problem. By the complex and real representations of the quaternion matrix, Jiang [4]and Wang [5] discussed the algebraic algorithms for QLS problem and derived someimportant theoretical results.

Note that the best approximate solution of the inconsistent equation (1.1) can beexpressed as the unique solution to the following consistent restricted equation

Ax = PAb, x ∈ Rr(

A∗) . (1.2)

Moreover, (1.2) can be generalized into the following restricted linear quaternionicsystems of equations

Ax = b, x ∈ T1, (1.3)

andx A = b, x ∈ T2, (1.4)

where Ti , i = 1, 2, are some special column and row spaces over H, respectively. Thus,investigating linear quaternionic systems of equations under some restrictions cameto be important and interesting. It provides us a new way to compute the approximatesolution to (1.1). In addition, suppose that (1.3) has an unique solution, then it canbe expressed as x = A(2)

T1,S1b, where S1 is arbitrary such that AT1 ⊕ S1 = H

n . It

follows that the generalized inverse A(2)T,S plays an important role in expressing the

solutions to the restricted quaternionic equations. In [6,7], we derived some necessaryand sufficient conditions for the existences of A(2)

T,S over the quaternion skew field asfollows.

Lemma 1.1 (1) Suppose that A ∈ Hm×nr , T1 is a subspace of H

n×1 of dimensions ≤ r and S1 is a subspace of H

m×1 of dimension m − s. Then A has a {2}-inverseX such that Rr (X) = T1, Nr (X) = S1 if and only if

AT1 ⊕ S1 = Hm×1, AT1 is denoted as AT1 = {Ax, x ∈ T1},

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Cramer’s rules for various solutions

in which case X is unique and denoted by A(2)rT1,S1

.

(2) Suppose that A ∈ Hm×nr , T2 is a subspace of H

1×m of dimension t ≤ r and S2is a subspace of H

1×n of dimension n − t. Then A has a {2}-inverse X such thatRl (X) = T2, Nl (X) = S2 if and only if

T2 A ⊕ S2 = H1×n, T2 A is denoted as T2 A = {x A, x ∈ T2},

in which case X is unique and denoted by A(2)lT2,S2

.

By computing the direct sum of some quaternionic vector spaces, we can derivesome conditions for the existences of generalized inverse A(2)

T,S . Unfortunately, themultiplication of quaternions is not commutative, then it is a hard work for us to checkAT1 ∩ S1 = 0 (T2 A ∩ S2 = 0) satisfied or not, which is saying that the given resultsare not practical. Therefore, it is urgent to search some new conditions to ensure theexistences of the generalized inverse A(2)

T,S over the quaternion skew field.Cramer’s rule is often used as a basic method to solve the unique solution to some

consistent system or the best approximate solution to some inconsistent system. How-ever, by some existing definitions of the determinant of quaternion matrix (such asthe Dieudonne and Chen), we can not get the classical adjoint matrix or its analogueeither. Therefore, we cannot derive the Cramer’s rule for some quaternion equations.Kyrchei [8,9] defined the row and column determinants of a square matrix over thequaternion skew field and derived some Cramer’s rules for quaternion equations. Byhis results, any component of the unique solution can be derived by the quotient of therow an column determinants of some square matrices of big orders. In this paper, wemainly consider some concise determinantal expressions of the generalized inverseA(2)

T,S, as well as some condensed Cramer’s rules for the general solutions, the leastsquares solutions and the best approximate solution to the systems (1.3)–(1.4).

The paper is organized as follows. In Sect. 2, we start with some basic concepts andresults about the row and column determinants of a square matrix over the quaternionskew field. In Sect. 3, by checking the nonsingularity of some special matrices weshow some new necessary and sufficient conditions for the existence of the generalizedinverse A(2)

T,S over the quaternion skew filed. Moreover, we show a series of concisedeterminantal expressions for these generalized inverses. In Sect. 4, we derive somecondensed Cramer’s rules for the general solutions, the least squares solutions as wellas the best approximate solution of the restricted quaternionic systems (1.3) and (1.4),respectively. Some results are even new for the complex cases. In Sect. 5, we show anumerical example to illustrate the main results. To conclude this paper, in Sect. 6 wepropose some further research topics.

2 Preliminaries

The definition of the determinant of a square matrix plays a key role in representingthe solution of a system of linear equations. Unlike multiplication of real or complexnumbers, multiplication of quaternions is not commutative. Many authors [10–14] hadtried to give the definitions of the determinants of a quaternion matrix. Unfortunately,

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by their definitions it is impossible for us to give a determinantal representation ofan inverse matrix. In 1991, Chen [15,16] offered a new definition and obtained thedeterminant representation of an inverse matrix over the quaternion skew field. How-ever this determinant also cannot be expanded by cofactors along an arbitrary row orcolumn with the exception of the nth row. Therefore, he had not obtained the classicaladjoint matrix or its analogue either. In 2008, I.I. Kyrchei defined the row and columndeterminants of a square matrix over the quaternion skew field as follows.

Suppose Sn is the symmetric group on the set In = {1, . . . , n}.Definition 2.1 (Definition 2.4 [8]) The i th row determinant of A = (

ai j) ∈ H

n×n isdefined by

rdeti A =∑

σ∈Sn

(−1)n−r aiik1aik1 ik1+1

· · · aik1+l1 i · · · aikr ikr +1 · · · aikr +lr ikr

for all i = 1, . . . , n. The elements of the permutation σ are indices of each monomial.The left-ordered cycle notation of the permutation σ is written as follows:

σ = (i ik1 ik1+1 · · · ik1+l1

) (ik2 ik2+1 · · · ik2+l2

) · · · (ikr ikr +1 · · · ikr +lr

).

The index i opens the first cycle from the left and other cycles satisfy the followingconditions, ik2 < ik3 < · · · < ikr and ikt < ikt +s for all t = 2, . . . , r and s = 1, . . . , lt .

Definition 2.2 (Definition 2.5 [8]) The j th column determinant of A = (ai j

) ∈ Hn×n

is defined by

cdet j A =∑

τ∈Sn

(−1)n−r a jkr jkr +lr· · · a jkr +1 jkr

· · · a j jk1+l1· · · a jk1+1 jk1

a jk1 j

for all j = 1, . . . , n. The elements of the permutation τ are indices of each monomial.The right-ordered cycle notation of the permutation τ is written as follows:

τ = (jkr +lr · · · jkr +1 jkr

) (jk2+l2 · · · jk2+1 jk2

) · · · ( jk1+l1 · · · jk1+1 jk1 j).

The index j opens the first cycle from the right and other cycles satisfy the followingconditions, jk2 < jk3 < · · · < jkr and jkt < jkt +s for all t = 2, . . . , r and s =1, . . . , lt .

Suppose that A. j (b) denotes the matrix obtained from A by replacing its j th columnwith the column b, and Ai.(b) denotes the matrix obtained form A by replacing its i throw with the row b.

Theorem 2.1 (Theorem 3.1 [8]) If a matrix A = (ai j

) ∈ Hn×n is Hermitian, then

rdet1 A = · · · rdetn A = cdet1 A = · · · cdetn A ∈ R.

Thus we define the determinant of a Hermitian matrix by putting det A = rdeti A =cdeti A for all i = 1, . . . , n. The following theorem about determinantal representationof an inverse matrix of Hermitian follows immediately:

123


Theorem 2.2 (Theorem 5.1 [8]) If a Hermitian matrix A = (ai j

) ∈ Hn×n is such that

det A �= 0, then there exist a unique right inverse matrix (R A)−1 and a unique leftinverse matrix (L A)−1 , and (R A)−1 = (L A)−1 =: A−1. They possess the followingdeterminantal representations:

(R A)−1 = 1

det A

⎡

⎢⎢⎢⎣

R11 R21 · · · Rn1R12 R22 · · · Rn2...

.... . .

...

R1n R2n · · · Rnn

⎤

⎥⎥⎥⎦

, (L A)−1 = 1

det A

⎡

⎢⎢⎢⎣

L11 L21 · · · Ln1L12 L22 · · · Ln2...

.... . .

...

L1n L2n · · · Lnn

⎤

⎥⎥⎥⎦

.

Here Ri j , Li j are right and left i j -th cofactors of A respectively for all i, j = 1, . . . , n.

Definition 2.3 (Definition 7.2 in [8]) For A ∈ Hn×n, the determinant of its corre-

sponding Hermitian matrix is called its double determinant, i.e.

ddet A := det(

A∗ A) = det

(AA∗) .

Theorem 2.3 (Theorem 8.1 in [8]) A necessary and sufficient condition of invertibilityof A ∈ H

n×n is ddet A �= 0. Then there exist A−1 = (L A)−1 = (R A)−1 , where

(L A)−1 = (A∗ A

)−1A∗ = 1

ddet A

⎡

⎢⎢⎢⎣

L11 L21 · · · Ln1L12 L22 · · · Ln2...

.... . .

...

L1n L2n · · · Lnn

⎤

⎥⎥⎥⎦

,

(R A)−1 = A∗ (AA∗)−1 = 1

ddet A

⎡

⎢⎢⎢⎣

R11 R21 · · · Rn1R12 R22 · · · Rn2...

.... . .

...

R1n R2n · · · Rnn

⎤

⎥⎥⎥⎦

,

and Li j = cdet j (A∗ A). j(a∗.i

), Ri j = rdeti (AA∗)i.

(a∗

j.

)for all i, j = 1, . . . , n.

3 Determinantal representations of the generalized inverses A(2)T,S

over the quaternion skew field

Determinantal representations of some generalized inverses are useful for theoreticalanalysis (e.g. [17–21]). Kyrchei [22,23] derived some determinantal representationsof the Moore–Penrose inverse over the quaternion skew field. Note that the generalizedinverse A(2)

T,S plays an important role in solving the solutions to some restricted systemof equations. Moreover, the Moore–Penrose inverse, the Drazin inverse and the Bott–Duffin inverse are all special cases of the generalized inverse A(2)

T,S . Thus it is important

and useful to study the determinantal representations of the generalized inverses A(2)T,S .

We begin with the following definitions.

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G.-J. Song et al.

Definition 3.1 (1) Two subspaces L and M of Hm are called complementary if H

m =L ⊕ M. In this case, every x ∈ H

m can be expressed uniquely as a sum x = y + z(y ∈ L , z ∈ M), we shall call y the R-projection of x on L along M.

(2) Two subspaces S and T of H1×n are called complementary if H

1×n = T ⊕ S.

In this case, every x ∈ H1×n can be expressed uniquely as a sum x = y + z

(y ∈ T, z ∈ S), we shall call y the L-projection of x on T along S.

Definition 3.2 (1) Let PL ,M denote the transformation that carries any x ∈ Hm into

projection on L along M. We denote PL ,M as both the linear transformation andits representation matrix. The linear transformation PL ,M is called the R-projectoron L along M and PL ,M x = y.

(2) Let QS,T denote the transformation that carries any x ∈ H1×m into projection on

T along S. We denote QS,T as both the linear transformation and its representationmatrix. The linear transformation QS,T is called the L-projector on T along Sand x QS,T = y.

In order to derive some concise determinantal expressions of the generalized inverseA(2)

T,S, as well as the conditions for its existence over the quaternion skew field, weneed the followings.

Definition 3.3 (1) Let A ∈ Hn×n and L1 be subspace of H

n . If APL1 + PL⊥1

isnonsingular, then the R-Bott–Duffin inverse of A with respect to L1, denote byA(−1)

r(L1), is defined by

A(−1)r(L1)

= PL1

(APL1 + PL⊥

1

)−1.

(2) Let A ∈ Hn×n and L2 be subspace of H

1×n . If QL2 A + QL⊥2

is nonsingular, then

the L-Bott–Duffin inverse of A with respect to L2, denote by A(−1)l(L2)

, is defined by

A(−1)l(L2)

=(

QL2 A + QL⊥2

)−1QL2 .

Lemma 3.1 (1) Suppose that A ∈ Hm×n, T1 is a subspace of H

n, C1 is a full rowrank matrix such that Nr (C1) = T1. Then the following results are equivalent.

(1a) T1 ∩ Nr (A) = 0;(1b) (A∗ A)(−1)

r(T1)exists and A∗ AT1 ⊕ T ⊥

1 = Hn;

(1c)

[A∗ A C∗

1C1 0

]is nonsingular;

(1d) A∗ A + C∗1 C1 is nonsingular.

(2) Suppose that A ∈ Hm×n, T2 is a subspace of H

1×m, C2 is a full column rankmatrix such that Nl (C2) = T2. Then the following results are equivalent.

(2a) T2 ∩ Nl (A) = 0;(2b) (AA∗)(−1)

l(T2)exists and T2 AA∗ ⊕ T ⊥

2 = H1×m;

(2c)

[AA∗ C2C∗

2 0

]is nonsingular;

123


(2d) AA∗ + C2C∗2 is nonsingular.

Proof For (1) (a)⇒(b). Note that T1 ∩ Nr (A) = 0, then dim AT1 = dim T1 andr(

APT1

) = r(PT1

). It follows that

(APT1

)∗ (APT1

) = T1 and r(PT1 A∗ APT1

) =r(PT1

). A∗ APT1 + PT ⊥

1is nonsingular then (A∗ A)(−1)

r(T1)exist and A∗ AT1 ⊕T ⊥

1 = Hn .

(b)⇒(c). Note that Nr (C) = T1 and (A∗ A)(−1)r(T1)

exists, therefore,

A∗ A(

A∗ A)(−1)

r(T1)= PA∗ AT1,T ⊥

1, I − A∗ A

(A∗ A

)(−1)

r(T1)= PT ⊥

1 ,A∗ AT1.

Rr

(C†

1

)= Rr

(C∗

1

) = T ⊥1 and C1 is of full row rank, then

C∗1

(C∗

1

)† = PRr(C∗1)

= PT ⊥1

, C1(

A∗ A)(−1)

r(T1)= 0

and

C1

(I − (

A∗ A)(−1)

r(T1)

(A∗ A

))C†

1 = C1C†1 = I.

Note that

A∗ A(

A∗ A)(−1)

r(T1)+ C∗

1

(C∗

1

)†(

I − A∗ A(

A∗ A)(−1)

r(T1)

)

= PA∗ AT1,T ⊥1

+ PT ⊥1

PT ⊥1 ,A∗ AT1

= I

and

A∗ A

(I − (

A∗ A)(−1)

r(T1)A∗ A

)C†

1 − C∗1

(C∗

1

)†(

I − A∗ A(

A∗ A)(−1)

r(T1)

)A∗ AC†

1

=(

I − A∗ A(

A∗ A)(−1)

r(T1)

)A∗ AC†

1 −(

I − A∗ A(

A∗ A)(−1)

r(T1)

)A∗ AC†

1 = 0,

thus

[A∗ A C∗

1C1 0

]−1

=⎡

⎣(A∗ A)(−1)

r(T1)

(I − (A∗ A)(−1)

r(T1)A∗ A

)C†

1(C†

1

)∗ (I − A∗ A (A∗ A)(−1)

r(T1)

) (C†

1

)∗A∗ A

((A∗ A)(−1)

r(T1)A∗ A − I

)C†

1

⎤

⎦ .

(c)⇒(d) is obviously. (d)⇒(a). Suppose that A∗ A + C∗1 C1 is nonsingular, then

[A

C1

]

is of full column rank and Nr

(A

C1

)= 0. It follows that T1 ∩ Nr (A) = 0.

Similarly, we can get the equivalences of (2a)–(2d). �

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G.-J. Song et al.

We can now prove the following results.

Theorem 3.2 (1) Suppose that A ∈ Hm×n, T1 is a subspace of H

n, S1 is a subspaceof H

m and dim (T1) = dim(S⊥

1

). In addition, B1 and C∗

1 are full column rankmatrices satisfy T1 = Nr (C1) , S1 = Rr (B1) . Then the generalized inverseA(2)

rT1,S1exists if and only if one of the following conditions is satisfied.

(1a)

[A B1

C1 0

]is nonsingular;

(1b)(

A∗ PS⊥1

A)−1

r(T1)

or(

APT1 A∗)−1r(S⊥

1 )exists;

(1c)

[A∗ PS⊥

1A C∗

1

C1 0

]or

[APT1 A∗ B1

B∗1 0

]is nonsingular;

(1d) A∗ PS⊥1

A + C∗1 C1 or APT1 A∗ + B1 B∗

1 is nonsingular.


1×m, S2 is a subspace of H1×n

and dim (T2) = dim(S⊥

2

). In addition, B2 and C∗

2 are full row rank matrices

satisfy T2 = Nl (C2) , S2 = Rl (B2) . Then the generalized inverse A(2)lT2,S2

exists

if and only if one of the following conditions is satisfied.

(2a)

[A B2

C2 0

]is nonsingular;

(2b)(

AQS⊥2

A∗)−1

l(T1)

or(

A∗QT2 A)−1

l(S⊥

2 )exists;

(2c)

[AQS⊥

2A∗ C2

C∗2 0

]or

[AQT2 A∗ B2

B∗2 0

]is nonsingular;

(2d) AQS⊥2

A∗ + C2C∗2 or A∗QT2 A + B∗

2 B2 is nonsingular.

Proof For (1a). Suppose that A(2)rT1,S1

exists, and by the definition of A(2)rT1,S1

we have

AA(2)rT1,S1

= PAT1,S1 , I − AA(2)rT1,S1

= PS1,AT1 ,

and

B1 B†1 = PRB1

= PS1 , C1 A(2)rT1,S1

= 0.

Noting that Rr

(C†

1

)= Rr

(C∗

1

) = T ⊥1 and C1 is of full row rank then

C1

(I − A(2)

rT1,S1A)

C†1 = C1C†

1 = I.

Moreover,

AA(2)rT1,S1

+ B1 B†1

(I − AA(2)

rT1,S1

)= PAT1,S1 + PS1 PS1,AT1 = PAT1,S1 + PS1,AT1 = I

123


and

A(

I − A(2)rT1,S1

A)

C†1 − B1 B†

1

(I − AA(2)

rT1,S1

)AC†

1 =(

I − AA(2)rT1,S1

)AC†

1

−(

I − AA(2)rT1,S1

)AC†

1 = 0,

then by a direct computation we have

[A B1

C1 0

]−1

=⎡

⎣A(2)

rT1,S1(I − A(2)

rT1,S1A)C†

1

B†1

(I − AA(2)

rT1,S1

)B†

1

(AA(2)

rT1,S1A − A

)C†

⎤

⎦ .

For (1b)–(1d). Suppose that A(2)rT1,S1

exists, it follows from Theorem 1.1 that AT1⊕S1 =H

m which is equivalent to T1 ∩Nr (A) = 0. Since Nr

(A∗ PS⊥

1A)

= Nr

(PS⊥

1A)

and

r(

PS⊥1

A)

= r (A) , we have Nr

(A∗ PS⊥

1A)

= Nr (A) , thus T1 ∩ Nr

(A∗ PS⊥

1A)

=0. By the Proof of Lemma 3.1, we can get (1b)–(1d), respectively.

Similarly, we can get the equivalences of (2a)–(2d). �

Theorem 3.3 (1) Let A, T1, S1, B1 and C1 be defined as Theorem 3.2 (1). Supposethat A(2)

rT1,S1exists, and dim (T1) = dim

(S⊥

1

) = p1. Denote

EB1 = I −B1 B†1 , FC1 = I −C†

1C1, M1 =[

A∗EB1 A C∗1

C1 0

], M2 =

[AFC1 A∗ B1

B∗1 0

],

then(a)

[A(2)

rT1,S1

0

]

=[

A∗EB1 A C∗1

C1 0

]−1 [A∗EB1

0

]

and A(2)rT1,S1

= (ai j

) ∈ Hn×m possess the following determinantal representations:

ai j = cdeti (M1).i(u. j

)

det M1(3.1)

where u. j is the j-th column of

[A∗EB1

0

]for all i = 1, . . . , n, j = 1, . . . , m.

(b)

[A(2)

rT1,S10]

= [FC1 A∗ 0

] [A∗FC1 A B1

B∗1 0

]−1

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G.-J. Song et al.

and A(2)rT1,S1

= (ai j


ai j = rdet j (M2) j. (ui.)

det M2(3.2)

where ui. is the i-th row of[

FC1 A∗ 0]

for all i = 1, . . . , n, j = 1, . . . , m.

(c)

A(2)rT2,S2

= (A∗EB1 A

)(−1)

r(T2)A∗EB1 = (

A∗EB1 A + C∗1 C1

)−1A∗EB1,

and A(2)rT1,S1

= (ai j


ai j = cdeti(

A∗EB1 A + C∗1 C1

).i

(u. j

)

det(

A∗EB1 A + C∗1 C1

) (3.3)

where u j. is the j-th column of A∗EB1 for all i = 1, . . . , n, j = 1, . . . , m.

(d)

A(2)rT1,S1

= FC1 A∗ (AFC1 A∗)(−1)

r(T1)= FC1 A∗ (

AFC1 A∗ + B1 B∗1

)−1,

and A(2)rT1,S1

= (ai j


ai j =rdet j

(AFC1 A∗ + B1 B∗

1

)j. (ui.)

det(

AFC1 A∗ + B1 B∗1

) (3.4)

where ui. is the i-th row of FC1 A∗ for all i = 1, . . . , n, j = 1, . . . , m.

(2) Let A, T2, S2, B2 and C2 be defined as Theorem 3.2 (2). Suppose that A(2)lT2,S2

exists

and dim (T1) = dim(S⊥

1

) = q1. Denote

EB2 = I −B2 B†2 , FC2 = I −C†

2C2, N1 =[

A∗EB2 A C∗2

C2 0

], N2 =

[A∗FC2 A B2

B∗2 0

],

then(a)

[A(2)

lT2,S2

0

]

=[

A∗EB2 A C∗2

C2 0

]−1 [A∗EB2

0

],

and A(2)lT2,S2

= (ai j


ai j = cdeti N1.i(v. j

)

det N1

123


where v. j is the j-th column of

[A∗EB2

0

]for all i = 1, . . . , n, j = 1, . . . , m.

(b)

[A(2)

lT2,S20]

= [FC2 A∗ 0

] [A∗FC2 A B2

B∗2 0

]−1

and A(2)lT2,S2

= (ai j


ai j = rdet j N2 (vi.)

det N2

where vi. is the i-th row of[

FC2 A∗ 0]

for all i = 1, . . . , n, j = 1, . . . , m.

(c)

A(2)lT2,S2

= (A∗EB2 A

)(−1)

l(T2)A∗EB2 = (

A∗EB2 A + C∗2 C2

)−1A∗EB1 ,

and A(2)lT2,S2

= (ai j


ai j = cdeti(

A∗EB2 A + C∗2 C2

).i

(v. j

)

det(

A∗EB2 A + C∗2 C2

)

where vi. is the i-th row of A∗EB1 for all i = 1, . . . , n, j = 1, . . . , m.

(d)

A(2)lT1,S1

= FC2 A∗ (AFC2 A∗)(−1)

l(T2)= FC2 A∗ (

AFC2 A∗ + B2 B∗2

)−1.

and A(2)lT2,S2

= (ai j


ai j = rdet j(

AFC2 A∗ + B2 B∗2

)(vi.)

det(

AFC2 A∗ + B2 B∗2

)

where vi. is the i-th column of FC2 A∗ for all i = 1, . . . , n, j = 1, . . . , m.

Proof (1a) Since Rr

(A(2)

rT1,S1

)= T1, Nr

(A(2)

rT1,S1

)= S1, we have C1 A(2)

rT1,S1= 0.

Note that EB1 = PS⊥1, then A∗EB1 AA(2)

rT1,S1= A∗EB1 . It follows that

[A∗EB1 A C∗

1C1 0

] [A(2)

rT1,S1

0

]

=[

A∗EB1 AA(2)rT1,S1

C1 A(2)rT1,S1

]

=[

A∗EB1

0

].

123

G.-J. Song et al.

Denote

[A∗EB1

0

]= (

ui j)

and by Theorem 2.2, we have

[A(2)

rT1,S1

0

]

=[

A∗EB1 A C∗1

C1 0

]−1 [A∗EB1

0

]

= 1

det M1

⎡

⎢⎢⎢⎣

L11 L21 · · · L(n+p)1L12 L22 · · · L(n+p)2...

.... . .

...

L1(n+p) L2(n+p) · · · L(n+p)(n+p)

⎤

⎥⎥⎥⎦

[A∗EB1

0

]

= 1

det M1

⎡

⎢⎢⎢⎣

∑k Lk1uk1

∑k Lk1uk2 · · · ∑

k Lk1ukm∑k Lk2uk1

∑k Lk2uk2 · · · ∑

k Lk2ukm...

.... . .

...∑k Lk(n+p)uk1

∑k Lk(n+p)uk2 · · · ∑

k Lk(n+p)ukm

⎤

⎥⎥⎥⎦

.

By the virtue of

∑

k

Lki mkj = cdeti

[A∗EB1 A C∗

1C1 0

]

.i

(u. j

)

where u. j is the j-th column of

[A∗EB2

0

]for all i = 1, . . . , n + p, j = 1, . . . , m,

we can derive (3.1).(1b) Similar to the proof of (1a), we can get A(2)

rT1,S1B1 = 0 and A(2)

rT1,S1AFC1 A∗ =

FC1 A∗, then

[A(2)

rT1,S10] [

A∗FC1 A B1B∗

1 0

]=

[A(2)

rT1,S1A∗FC1 A A(2)

rT1,S1B1

]= [

FC1 A 0].

Recall Theorem 2.2, we have

[A(2)

rT1,S10]

= [FC1 A 0

] [A∗FC1 A B1

B∗1 0

]−1

= 1

det M2

[FC1 A 0

]

⎡

⎢⎢⎢⎣

R11 R21 · · · R(n+p)1R12 R22 · · · R(n+p)2...

.... . .

...

R1(n+p) R2(n+p) · · · R(n+p)(n+p)

⎤

⎥⎥⎥⎦

= 1

det M2

⎡

⎢⎢⎢⎣

∑k v1k R1k

∑k v1k R2k · · · ∑

k v1k R(n+p)k∑k v2k R1k

∑k v1k R2k · · · ∑

k v2k R(n+p)k...

.... . .

...∑k vmk R1k

∑k vmk R2k · · · ∑

k vmk R(n+p)k

⎤

⎥⎥⎥⎦

.

123


By the virtue of

∑

k

v jk Rik = rdeti

[A∗EB1 A C∗

1C1 0

]

i.

(v j.

)

where v j. is the j-th row of[

FC1 A 0]

for all i = 1, . . . , n + p, j = 1, . . . , m,

then we can derive (3.2).(1c) By the proof of Theorem 3.2 and note that

(C†

1

)∗A∗EB1 A

((A∗EB1 A

)(−1)

r(T1)A∗EB1 A − I

)C†

1 = 0,

we have

[A∗EB1 A C∗

1C1 0

]−1

=

⎡

⎢⎢⎣

(A∗EB1 A

)(−1)

r(T1)

(I − (

A∗EB1 A)(−1)

r(T1)A∗EB1 A

)C†

1(

C†1

)∗ (I − A∗EB1 A

(A∗EB1 A

)(−1)

r(T1)

)0

⎤

⎥⎥⎦,

thus A(2)rT1,S1

= (A∗EB1 A

)(−1)

r(T1)A∗EB1 . By the proof of (1), we can get (3.3).

(1d) Since

(A∗EB1 A + C∗

1 C1)

A(2)rT1,S1

= A∗EB1 AA(2)rT1,S1

+ C∗1 C1 A(2)

rT1,S1= A∗EB1 ,

we have

A(2)rT1,S1

= (A∗EB1 A + C∗

1 C1)−1

A∗EB1 .

Consequently, (3.4) is satisfied. Similarly, we can get (2a)–(2d). �Remark 3.1 The Moore–Penrose inverse A†, the Weighted Moore–Penrose inverseA†

M,N and the Drazin inverse AD are all special cases of the generalized inverses

A(2)rT1,S1

and A(2)lT2,S2

, then we can get their determinantal representations by Theorem3.3, respectively.

4 Cramer’s rule for the general solutions to some restricted quaternionicsystems of linear equations

In 1970, Steve Robinson [24] gave an elegant proof of Cramer’s rule over the complexnumber field: rewriting

Ax = b (4.1)

123

G.-J. Song et al.

as

A · I (i → x) = A (i → b) ,

where I is an identity matrix of order n, and taking determinants

det (A) · det I (i → x) = det (A (i → b)) .

Since det I (i → x) = xi , i = 1, . . . , n, it follows that

xi = det (A (i → b))

det (A), i = 1, . . . , n,

which is called Cramer rule. A Cramer’s rule for the minimum-norm least-squaressolution of the linear system (4.1) with a general rectangular matrix was obtained in1982 by Ben-Israel [25] and Verghese [26]. Since then, the research on Cramer’s rulehas been very active and is mainly focused either on the extension to various otherlinear systems or on more condensed form of the rule for the linear system (e.g. [27–33]). In particular, Cramer’s rule for some left and right quaternionic systems of linearequations was given by Kyrchei in [31] as follows.

Lemma 4.1 (1) LetAx = y (4.2)

be a right system of linear equations with a matrix of coefficients A ∈ Hm×n,

a column of constants y = [y1, . . . , yn

]T ∈ Hn, and a column of unknowns

x = [x1, . . . , xn

]T. If ddet A = det (A∗ A) �= 0, then the solution to linear

system (4.3) is given by the components

x j = cdet j (A∗ A). j ( f )

ddet A, j = 1, . . . , n,

where f = A∗y.(2) Let

x A = y (4.3)

be a left system of linear equations with a matrix of coefficients A ∈ Hm×n, a

column of constants y = [y1, . . . , yn

] ∈ H1×m, and a column of unknowns

x = [x1, . . . , xn

]. If ddet A = det (AA∗) �= 0, then the solution to linear

system (4.3) is given by the components

xi = rdeti (AA∗)i. ( f )

ddet A, i = 1, . . . , n,

where f = y A∗.

123


In this section, we aim to consider some Cramer’s rules for various solutions to therestricted quaternionic systems of linear equations

Ax = b, x ∈ T1, (4.4)

andx A = b, x ∈ T2, (4.5)

respectively.

Lemma 4.2 (1a) The system (4.4) is consistent if and only if b ∈ AT1, and thegeneral solutions can be expressed as

x = (APT1

)†b +

(I − (

APT1

)†A)

PT1 z1, (4.6)

where z1 is arbitrary.

(1b) The system (4.4) has a unique solution if and only if b ∈ AT1 and T1 ∩Nr (A) ={0} . Then the unique solution can be expressed as

x = A(2)rT1,S1

b,

where S1 is arbitrary such that AT1 ⊕ S1 = Hm .

(2a) The system (4.5) is consistent if and only if b ∈ T2 A, and the general solutionscan be expressed as

x = b(QT2 A

)† + z2 QT2

(I − A

(QT2 A

)†)

,

where z2 is arbitrary.(2b) The system (4.5) has unique solution if and only if b ∈ T2 A and T2∩Nl (A) = {0} .

Then the unique solution can be expressed as

x = bA(2)lT2,S2

,

where S2 is arbitrary such that T2 A ⊕ S2 = H1×n .

Proof (1a) It is easy to verify x defined as (4.6) is a solution to the restricted system(4.4). Conversely, for any x0 satisfies the Eq. (4.4), there exist y0 such thatx0 = PT1 y0. Therefore, x0 can be written as

x0 = (APT1

)†b +

(I − (

APT1

)†A)

PT1 y0,

which is saying that (4.6) is the general solutions to the restricted system (4.4).

123

G.-J. Song et al.

(1b) If b ∈ AT1 then it is obviously that the system (4.4) has a solution x0 ∈ T1. Letthe general solutions of the system (4.4) be x = x0 + y ∈ T1, where y ∈ Nr (A) .

Then y = x − x0 ∈ T1. Note that T1 ∩ Nr (A) = {0} then y = 0. Therefore,(4.4) has a unique solution x = x0. Conversely, let the general solutions of (4.4)be x = x0 + y ∈ T1, where y ∈ Nr (A) and x0 ∈ T1 be a particular solution of(4.4). Since the system (4.4) has a unique solution thus y = 0. Moreover, x = x0and b = Ax0 ∈ AT1. It follows from y ∈ Nr (A) , y = x − x0 ∈ T1 and y = 0that T1 ∩ Nr (A) = {0} . Moreover, since x = A(2)

rT1,S1b ∈ T1, AA(2)

rT1,S1is the

projector PAT1,S1 and b ∈ AT1, then we have Ax = AA(2)rT1,S1

b = b. Therefore,

x = A(2)rT1,S1

b is the solution to (4.4). Similarly, we can get (2a)–(2b). �We can now prove the following results.

Theorem 4.3 (1) Let A, B1, C1 and T1 be defined as Theorem 3.2 (1). Denote M1 =[A B1

C1 0

], M2 =

[A∗EB1 A C∗

1C1 0

]. Suppose that b ∈ AT1 and T1 ∩ Nr (A) =

{0} , then (4.4) has a unique solution x0 = (xi ) , which can be expressed as(a)

x j =cdet j

(M∗

1 M). j ( f )

det(M∗

1 M1) , j = 1, . . . , n,

where f = M∗1 b.

(b)

x j = cdet j (M2). j ( f )

det M2, j = 1, . . . , n,

where f =[

A∗EB1 b0

].

(c)

x j =cdet j

(A∗EB2 A + C∗

2 C2). j ( f )

det(

A∗EB2 A + C∗2 C2

) , j = 1, . . . , n,

where f = A∗EB1 b.

(2) Let A, B2, C2 and T2 be defined as Theorem 3.2 (2). Denote N1 =[

A B2C2 0

], N2 =

[A∗EB2 A C∗

2C2 0

]. Suppose that b ∈ T2 A and T2 ∩ Nl (A) = {0} , then (4.5) has

a unique solution x0 = (xi ), which can be expressed as(a)

xi = rdeti(N1 N∗

1

)i.

( f )

det(N1 N∗

1

) , i = 1, . . . , m,

123


where f = bN∗1 .

(b)

xi = rdeti (N2)i. ( f )

det N2, i = 1, . . . , m,

where f = [bEB2 A∗ 0

].

(c)

xi = rdeti(

A∗EB2 A + C∗2 C2

)i. ( f )

det(

A∗EB2 A + C∗2 C2

) , i = 1, . . . , m,

where f = bEB2 A∗.

Proof (1) It follows the proof of Lemma 4.1 that the unique solution of (4.4) can beexpressed as A(2)

rT1,S1b. Then (1a)–(1c) follow from the Cramer’s rules of the generalized

inverse A(2)rT1,S1

, respectively. Similarly, we can get (2a)–(2c). �

If b ∈ AT1, and T1 ∩ Nr (A) �= {0} , then (4.4) is consistent, however, the solutionis not unique. In this case, for any subspaces S1, A(2)

rT1,S1does not exist, and we can

not investigate any solution to the system (4.4) by the Cramer’s rule of A(2)rT1,S1

. So as

to derive the Cramer’s rule for the general solutions to (4.4), we need construct a new

system. Setting�T1 = Rr

(PT1 A∗) , the restricted system

Ax = b, x ∈ �T1 (4.7)

is consistent and the solution is unique which can be expressed as x = A(2)r�

T1,�S1

b, where

�S1 is an arbitrary subspace of H

m such that�

AT1 ⊕ �S1 = H

m . Thus, we can derive thegeneral solutions to the system (4.4) by the Cramer’s rule of the unique solution to (4.7).

Moreover, in order to avoid adding a new variant�S1, choose

�S1 = (AT1)

⊥ = (AT1)⊥ ,

then A(2)r�

T1,�S1

= A(2)r�

T1,(AT1)⊥ . Similarly, if T2 ∩ Nl (A) �= {0} and b ∈ T2 A, then the

restricted equation

x A = b, x ∈ �T2 (4.8)

where�T2 = Rl

(A∗ PT2

), is consistent and the solution is unique which can be

expressed as x = bA(2)l�T2,

�S2

.

In order to show the main result of this section, we need the following results.

123

G.-J. Song et al.

Lemma 4.4 (1) Suppose that A ∈ Hm×n, T1 is a subspace of H

n . Denote�T1 =

Rr(PT1 A∗) then

(APT1

)† = (A∗ A)(−1)r(�

T1

) A∗, and

(APT1

)† = (A∗ A

)(−1)

r(T1)A∗ ⇔ T1 ∩ Nr (A) = {0} .


1×m . Denote�T2 = Rr

(AQT2

)

then(QT1 A

)† = A∗ (AA∗)(−1)l(�

T2

) , and

(QT1 A

)† = A∗ (AA∗)(−1)

l(T2)⇔ T2 ∩ Nl (A) = {0} .

Proof (1) Since�T1 ∩ Nr (A) = {0} , it follows from Lemma 3.1 that (A∗ A)(−1)

r(�T1

)

exists. Moreover, note that APT1 = AP�T1

then(

APT1

)† = (PT1 A∗ APT1

)†PT1 A∗ =

(A∗ A)(−1)r(�

T1

) A∗. If T1 ∩ Nr (A) = {0} , thus T1 = �T1 ⊕ T1 ∩ Nr (A) = �

T1, and

(APT1

)† = (A∗ A

)(−1)

r(T1)A∗. (4.9)

Conversely, suppose that (4.9) is satisfied, then (A∗ A)(−1)r(�

T1

) A∗ A = (A∗ A)(−1)r(T1)

A∗ A,

which is saying that P�T1,(A∗ AT1)

⊥ = PT1,(A∗ AT1)⊥ , then T1 = �

T1 ⇒ T1 ∩ Nr (A) ={0} . Similarly, we can get (2). �Lemma 4.5 (1) Let A ∈ H

m×n, T1 be a subspace of Hn and

�T1 = Rr

(PT1 A∗) .

Suppose that C1 is a full row rank matrix satisfies T1 = Nr (C1). If T1 ∩Nr (A) �={0} , then there exist a full column rank matrix D1 such that T1 ∩ Nr (A) =Rr

(D∗

1

). Then

�T1 = Nr

(C1D1

), C1 D∗

1 = 0,

G1 = A∗ A + C∗1 C1 + D∗

1 D1 and W1 =⎡

⎣A∗ A C∗

1 D∗1

C1 0 0D1 0 0

⎤

⎦

are nonsingular with

⎡

⎣A∗ A C∗

1 D∗1

C1 0 0D1 0 0

⎤

⎦

−1

=

⎡

⎢⎢⎢⎢⎣

(A∗ A)(−1)r(�

T1

) U1C†1 U1 D†

1

(U1C†

1

)∗ −(

C†1

)∗A∗ AU1C†

1 −(

C†1

)∗A∗ AU1 D†

1(U1 D†

1

)∗ −(

D†1

)∗A∗ AU1C†

1 −(

D†1

)∗A∗ AU1 D†

1

⎤

⎥⎥⎥⎥⎦

,

123


where

(A∗ A

)(−1)

r(�T1

) = G−11 − G−1

1

[C1D1

]∗ ([C1D1

]G−1

1

[C1D1

]∗)−1 [C1D1

]G−1

1

and

U1 =⎛

⎝I − (A∗ A

)(−1)

r(�T1

) A∗ A

⎞

⎠ .

(2) Let A ∈ Hm×n, T2 be a subspace of H

1×m and�T2 = Rl

(A∗QT2

). Suppose that

C2 is a full row rank matrix satisfies T2 = Nl (C2). If T2 ∩ Nl (A) �= {0} thenthere exist a full row rank matrix D2 such that T2 ∩ Nl (A) = Rl (D2) . Then�T2 = Nl

(C2 D2

), D∗

2C2 = 0,

G2 = AA∗ + C2C∗2 + D2 D∗

2 and W2 =⎡

⎣AA∗ C2 D2C∗

2 0 0D∗

2 0 0

⎤

⎦

are nonsingular with

⎡

⎣AA∗ C2 D2C∗

2 0 0D∗

2 0 0

⎤

⎦

−1

=

⎡

⎢⎢⎢⎢⎣

(AA∗)(−1)l(�

T2

)

(C†

2U2

)∗ (D†

2U2

)∗

C†2U2 −C†

2U2 AA∗(

C†2

)∗ −D†2U2 AA∗

(C†

2

)∗

D†2U2 −C†

2U2 AA∗(

D†2

)∗ −D†2U2 AA∗

(D†

2

)∗

⎤

⎥⎥⎥⎥⎦

(4.10)where

(A∗ A

)(−1)

l(�T2

) = G−12 − G−1

2

[C2 D2

] ([C∗

2D∗

2

]G−1

2

[C2 D2

])−1 [C∗

2D∗

2

]G−1

2 ,

and

U2 =⎛

⎝I − AA∗ (AA∗)(−1)

l(�T2

)

⎞

⎠ .

Proof (1) Since

�T1

⊥= T ⊥

1

⊥⊕ T1 ∩ Nr (A) = Rr (C1)⊥⊕ Rr

(D∗

1

) = Rr([

C∗1 D∗

1

]),

123

G.-J. Song et al.

we can get

C1 D∗1 = 0 and

�T1 = Nr

(C1D1

).

It follows that�T1 ∩ Nr (A) = {0} . Recall Lemma 3.1 we have W1 and A∗ A +

C∗1 C1 + D∗

1 D1 are nonsingular. Moreover,

⎡

⎣A∗ A + C∗

1 C1 + D∗1 D1 C∗

1 D∗1

C1 0 0D1 0 0

⎤

⎦

−1

=⎡

⎣A∗ A C∗

1 D∗1

C1 0 0D1 0 0

⎤

⎦

−1 ⎡

⎣I −C∗

1 −D∗1

0 I 00 0 I

⎤

⎦ ,

then by a direct computation we have (4.10). Similarly, we can get (2).�

We can now prove the Cramer’s rules for the general solutions to the restrictedquaternionic systems (4.4) and (4.5).

Theorem 4.6 (1) Let A1, T1,�T1, C1, D1 and W1 be defined as Lemma 4.5. Suppose

that

b ∈ AT1 and T1 ∩ Nr (A) = Rr(D∗

1

) �= {0} ,

then the general solutions of (4.4) can be expressed as(1a)

x j =cdet j (W1). j

⎛

⎝

⎡

⎣A∗b0

D1z

⎤

⎦

⎞

⎠

det W1, j = 1, . . . , n, z is arbi trary. (4.11)

(1b)

x j =cdet j

(A∗ A + C∗

1 C1 + D∗1 D1

). j

(A∗b + D∗

1 D1z)

det(

A∗ A + C∗1 C1 + D∗

1 D1) ,

j = 1, . . . , n, z is arbi trary. (4.12)

(2) Let A2, T2, C2, D2 and W2 be defined as Lemma 4.5. Suppose that

b ∈ T2 A and T2 ∩ Nl (A) = Rl (D2) �= {0} ,

then the general solutions of (4.5) can be expressed as:

123


(2a)

xi = rdeti (W2)i.([

bA∗ 0 zD2])

det W2, i = 1, . . . , n, z is arbi trary.

(2b)

xi =rdeti

(A∗ A + C∗

2 C2+D∗2 D2

)i.

(bA∗+zD2 D∗

2

)

det(

A∗ A + C∗2 C2 + D∗

2 D2) , i =1, . . . , n, z is arbi trary.

Proof (1a) It is easy to prove x0 = (A∗ A)(−1)r(�

T1

) A∗b is the unique solution to (4.7),

and the general solutions to the restricted equation (4.4) can be expressed as

x = (A∗ A

)(−1)

r(�T1

) A∗b + PT1∩Nr (A)z, (4.13)

where z is arbitrary. Note that

T1 = Nr (C) ,�T1 = Nr

[C1D1

], Ax0 = b, x0 ∈ �

T1,

thus

APT1∩Nr (A) = 0, D1 PT1∩Nr (A) = D1, C1 PT1∩Nr (A) = 0 and

[C1D1

]x0 = 0.

For x defined as (4.13), we have

A∗ Ax = A∗b, C1x = 0, D1x = D1z,

which can be written as

⎡

⎣A∗ A C∗

1 D∗1

C1 0 0D1 0 0

⎤

⎦

⎡

⎣x00

⎤

⎦ =⎡

⎣A∗b0

D1z

⎤

⎦ . (4.14)

It follows Theorem 4.3 (1b) that the Eq. (4.11) is satisfied.For (1b). Multiply (4.14) by

[I C∗

1 D∗1

]from left we have

(A∗ A + C∗

1 C1 + D∗1 D1

)x = A∗b + D∗

1 D1z.

Recall Theorem 4.3 (1c), we have (4.12). Similarly, we can get (2a)–(2b).�

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Under the assumption that b ∈ AT1, we can get Theorem 4.3 and 4.6. If b1 /∈ AT1,

then the system (4.4) is not consistent. In this case, we need consider the general leastsquares solutions to (4.4).

Theorem 4.7 (1) Suppose that b /∈ AT1, then (4.4) is not consistent. Denote�T1 =

Rr(PT1 A∗) , and assume that C1 and D1 are full row rank matrices such that

T1 = Nr (C), T1 ∩ Nr (A) = Rr(D∗

1

). Then the general least squares solution

to (4.4) can be expressed as

x = (APT1

)†b +

(I − (

APT1

)†A)

PT1 z1.

In this case, denote PAT1 b = At0, t0 ∈ T1, then any least squares solution to(4.4) can be expressed as

(a)

x j =cdet j (W1). j

⎛

⎝

⎡

⎣A∗ At0

0D1z

⎤

⎦

⎞

⎠

det W1, j = 1, . . . , n, z is arbitrary. (4.15)

(b)

x j =cdet j

(A∗ A + C∗

1 C1 + D∗1 D1

). j

(A∗ At0 + D∗

1 D1z)

det(

A∗ A + C∗1 C1 + D∗

1 D1) ,

j = 1, . . . , n, z is arbitrary. (4.16)

(2) Suppose that b /∈ T2 A, then (4.5) is not consistent. Denote�T2 = Rl

(A∗QT2

),

and assume that C2 and D2 are full column rank matrices such that T2 = Nl (C2),T2 ∩ Nl (A) = Rl (D2) . Then the general least squares solution to (4.4) can beexpressed as

x = b(QT2 A

)† + z2 QT2

(I − A

(QT2 A

)†)

,

In this case, denote bQ AT2 = t0 A, t0 ∈ T2, then for any least squares solution to(4.5) can be expressed as

(a)

xi = cdeti (W2).i(

t0 AA∗ 0 zD2)

det W2, i = 1, . . . , n, z is arbitrary. (4.17)

(b)

xi = cdeti(

AA∗ + C2C∗2 + D2 D∗

2

).i

(t0 AA∗ + zD2 D∗

2

)

det(

AA∗ + C2C∗2 + D2 D∗

2

) ,

i = 1, . . . , n, z is arbitrary. (4.18)

123


Proof (1) Since b /∈ AT1, then (4.4) is not consistent. Note that

Ax − b = (Ax − PAT1 b

) + (PAT1 b − b

),

therefore, for any x ∈ T1, we have

‖Ax − b‖22 = ∥∥Ax − PAT1 b

∥∥22 + ∥∥PAT1 b − b

∥∥22 .

Consequently, ‖Ax − b‖ is minimal if and only if Ax = PAT1 b. Moreover, by theProof of Theorem 4.6 and denote PAT1 b = At0, t0 ∈ T1, we can get (4.15)–(4.16).Similarly, we can get (4.17)–(4.18). �

Corollary 4.8 (1) If the system (4.4) is not consistent, then(

APT1

)†b is the minimum

norm least squares solution to (4.4) which can be expressed as (4.15) and (4.16)by setting z = 0.

(2) If the system (4.5) is not consistent, then b(QT2 A

)†is the minimum norm least

squares solution to (4.5) which can be expressed as (4.17) and (4.18) by settingz = 0.

5 Example

Let us consider the restricted quaternionic linear system

Ax = b, x ∈ T, (5.1)

where

A =⎡

⎣1 i 10 1 01 i 1

⎤

⎦ , b =⎡

⎣jij

⎤

⎦ , x =⎡

⎣x1x2x3

⎤

⎦ , T = Rr

⎛

⎝

⎡

⎣1 1i 0j −1

⎤

⎦

⎞

⎠ .

By a direct computation, we have

Nr (A) = Rr

⎛

⎝

⎡

⎣10

−1

⎤

⎦

⎞

⎠ , T = Nr([

1 i − k 1]) = Nr (C) .

Now we can find the general solution of (5.1) by Theorem 4.6. Note that

b ∈ AT, T ∩ Nr (A) = Rr

⎛

⎝

⎡

⎣10

−1

⎤

⎦

⎞

⎠ = Rr(D∗) , A∗b =

⎡

⎣2 j

i − 2k2 j

⎤

⎦

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G.-J. Song et al.

then,

A∗ A + C∗C + D∗ D

=⎡

⎣2 2i 2

−2i 3 −2i2 2i 2

⎤

⎦ +⎡

⎣1 i − k 1

−i + k 2 −i + k1 i − k 1

⎤

⎦ +⎡

⎣1 0 −10 0 0

−1 0 1

⎤

⎦

=⎡

⎣4 3i − k 2

−3i + k 5 −3i + k2 3i − k 4

⎤

⎦ ,

andcdet

(A∗ A + C∗C + D∗ D

) = 20.

Setting z =⎡

⎣z1z2z3

⎤

⎦ , then

D∗ Dz =⎡

⎣1 0 −10 0 0

−1 0 1

⎤

⎦

⎡

⎣z1z2z3

⎤

⎦ =⎡

⎣z1 − z3

0z3 − z1

⎤

⎦ =⎡

⎣u0

−u

⎤

⎦ ,

where u = z1 − z3 is arbitrary. It follows Theorem 4.6 (1b) that

x1 = 1

20

(cdet1

(A∗ A + C∗C + D∗D

)).1

⎛

⎝A∗b +⎡

⎣u0

−u

⎤

⎦

⎞

⎠

= 1

20cdet1

⎛

⎝2 j + u 3i − k 2i − 2k 5 −3i + k2 j − u 3i − k 4

⎞

⎠

= 1

20{(20 − (−3i + k) (3i − k)) (2 j + u)

+ (2 (3i − k) − 4 (3i − k)) (i − 2k) + ((3i − k) (−3i + k) − 10) (2 j − u)}= 1 + j + u

2,

x2 = 1

20

(cdet2

(A∗ A + C∗C + D∗ D

)).2

(A∗b

)

= 1

20cdet2

⎛

⎝4 2 j + u 2

−3i + k i − 2k −3i + k2 2 j − u 4

⎞

⎠

= 1

20{(2 (−3i + k) − 4 (−3i + k)) (2 j + u) + 16 (i − 2k) − 4 (i − 2k)

(+2 (−3i + k) − 4 (−3i + k)) (2 j − u)}= i,

123


x3 = 1

20

(cdet3

(A∗ A + C∗C + D∗D

)).3

(A∗b

)

= 1

20cdet3

⎛

⎝4 3i − k 2 j + u

−3i + k 5 i − 2k2 3i − k 2 j − u

⎞

⎠

= 1

20{((3i − k) (−3i + k) − 5 × 2) (2 j + u) + 2 (3i − k) (i − 2k)

−4 (3i − k) (i − 2k) + (4 × 5 − (3i − k) (−3i + k)) (2 j − u)}= 1 + j − u

2.

Note that

Cx = C

⎡

⎣x1x2x3

⎤

⎦ = [1 i − k 1

]⎡

⎣1+ j+u

2i

1+ j−u2

⎤

⎦ = 1 + j + u

2− 1 − j − u

2= 0,

which is saying that x =⎡

⎣1+ j+u

2i

1+ j−u2

⎤

⎦ with an arbitrary u is a solution to (5.1).

6 Conclusion

In this paper, we derive some new determinantal representations of the generalizedinverse A(2)

rT1,S1and A(2)

lT2,S2over the quaternion skew field by the theory of the column

and row determinants, respectively. Moreover, we show a series of Cramer’s rules forthe general solutions as well as the least squares solutions to restricted quaternionicsystems of linear equations (1.3)–(1.4). In the end, we give a numerical example toillustrate the main result.

Motivated by the work in this paper, it would be of interest to investigate thedeterminantal representations of the least squares inverse A(1,3), the minimum norminverse A(1,4), as well as some Cramer’s rules for the general solutions of the followingrestricted quaternion matrix equations

AX B = D,Rr (X) ⊂ T1,Nr (X) ⊃ S1,

and

AX B = D,Rl (X) ⊂ T2,Nl (X) ⊃ S2.

We will show the results in the following papers.

Acknowledgments This research was supported by the National Natural Science Foundation of China(No: 11326066), the Doctoral Program of Shan Dong Provience (BS2013SF011), a Project of ShandongProvince Higher Educational Science and Technology Program (J14LI01), Scientific research foundation

123

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of Shanghai Finance University (SHFUKT13-08), Shanghai Municipal Education Commission ResearchFoundation (No: ZZGJD 12061) and Scientific Research Foundation of Shanghai University of EngineeringScience (No: A-0501-11-020).

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Cramer’s rules for various solutions to some restricted quaternionic linear systems