2. Preliminaries

(1) over the complex field and the expression for its general solution. Navarra et al. [6] derived a new necessary and sufficient condition for the existence and a new representation of (1) over the complex field and used the results to give a simple representation. Wang [7] considers the system (1) over the quaternion skew field and gets its solvability conditions and a representa-

Throughout the chapter, we denote the real number field by R, the set of all m � n matrices

<sup>2</sup> <sup>¼</sup> <sup>j</sup> <sup>2</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> ¼ �1; <sup>a</sup>0; <sup>a</sup>1; <sup>a</sup>2; <sup>a</sup><sup>3</sup> <sup>∈</sup> <sup>R</sup>

by H<sup>m</sup>�<sup>n</sup> and by H<sup>m</sup>�<sup>n</sup> <sup>r</sup> , and the set of matrices over H with a rank r. For A ∈ H<sup>n</sup>�<sup>m</sup>, the symbols A\* stands for the conjugate transpose (Hermitian adjoint) matrix of A. The matrix

Generalized inverses are useful tools used to solve matrix equations. The definitions of the Moore-Penrose inverse matrix have been extended to quaternion matrices as follows. The

The determinantal representation of the usual inverse is the matrix with the cofactors in the entries which suggests a direct method of finding of inverse and makes it applicable through Cramer's rule to systems of linear equations. The same is desirable for the generalized inverses. But there is not so unambiguous even for complex or real generalized inverses. Therefore, there are various determinantal representations of generalized inverses because of looking for their more applicable explicit expressions (see, e.g. [8]). Through the noncommutativity of the quaternion algebra, difficulties arise already in determining the quaternion

The understanding of the problem for determinantal representation of an inverse matrix as well as generalized inverses only now begins to be decided due to the theory of column-row determinants introduced in [17, 18]. Within the framework of the theory of column-row determinants, determinantal representations of various kinds of generalized inverses and (generalized inverses) solutions of quaternion matrix equations have been derived by the

The main goals of the chapter are deriving determinantal representations (analogs of the classical Cramer rule) of general solutions of the system (1) and its simpler cases over the

The chapter is organized as follows. In Section 2, we start with preliminaries introducing of row-column determinants and determinantal representations of the Moore-Penrose and Cramer's rule of the quaternion matrix equations, AXB=C. Determinantal representations of a partial solution (an analog of Cramer's rule) of the system (1) are derived in Section 3. In Section 4, we give Cramer's rules to special cases of (1) with 1 and 2 one-sided equations.

, is the unique matrix X∈ H<sup>n</sup>�<sup>m</sup> satisfying

tion of a general solution.

4 Matrix Theory-Applications and Theorems

over the quaternion algebra

determinant (see, e.g. [9–16]).

quaternion skew field.

Finally, the conclusion is drawn in Section 5.

∈ H<sup>n</sup>�<sup>n</sup> is Hermitian if A\*

Moore-Penrose inverse of A ∈ H<sup>m</sup>�<sup>n</sup>, denoted by A†

A ¼ aij

H ¼ a<sup>0</sup> þ a1i þ a2j þ a3kji

=A.

ð Þ<sup>1</sup> AXA <sup>¼</sup> <sup>A</sup>, 2ð ÞXAX <sup>¼</sup> <sup>X</sup>, 3ð Þð Þ AX <sup>∗</sup> <sup>¼</sup> AX, and 4ð Þð Þ XA <sup>∗</sup> <sup>¼</sup> XA.

author (see, e.g. [19–25]) and by other reseachers (see, e.g. [26–29]).

For <sup>A</sup> <sup>¼</sup> aij � �<sup>∈</sup> <sup>M</sup>ð Þ <sup>n</sup>; <sup>H</sup> , we define <sup>n</sup> row determinants and <sup>n</sup> column determinants as follows. Suppose Sn is the symmetric group on the set In ¼ f g 1;…; n .

Definition 2.1. The ith row determinant of <sup>A</sup> <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>m</sup> is defined for all i <sup>¼</sup> <sup>1</sup>, …, n by putting

$$\mathbf{rdet}\_{i}\mathbf{A} = \sum\_{\sigma \in S\_{n}} (-1)^{n-r} \left( a\_{\dot{i}\dot{i}\_{k\_{1}}} a\_{\dot{i}\_{k\_{1}}\dot{i}\_{k\_{1}+1}} \dots a\_{\dot{i}\_{k\_{1}+l\_{1}}\dot{i}\_{1}} \right) \dots \left( a\_{\dot{i}\_{l\_{r}}\dot{i}\_{k\_{r}+1}} \dots a\_{\dot{i}\_{l\_{r}+l\_{r}}\dot{i}\_{l\_{r}}} \right),$$
 
$$\sigma = (\dot{i}\_{k\_{1}}\dot{i}\_{k\_{1}+1} \dots \dot{i}\_{k\_{1}+l\_{1}})(\dot{i}\_{k\_{2}}\dot{i}\_{k\_{2}+1} \dots \dot{i}\_{k\_{2}+l\_{2}}) \dots (\dot{i}\_{k\_{r}}\dot{i}\_{k\_{r}+1} \dots \dot{i}\_{k\_{r}+l\_{r}}),$$

with conditions ik<sup>2</sup> < ik<sup>3</sup> < … < ikr and ikt < iktþ<sup>s</sup> for all t ¼ 2, …, r and all s ¼ 1, …, lt.

Definition 2.2. The jth column determinant of <sup>A</sup> <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>m</sup> is defined for all j <sup>¼</sup> <sup>1</sup>, …, n by putting

$$\mathbf{cdet}\_{j}\mathbf{A} = \sum\_{\pi \in S\_{n}} (-1)^{n-r} \left( a\_{j\_{k}j\_{k\_{r}+l\_{r}}} \dots a\_{j\_{k\_{r}+1}i\_{k\_{r}}} \right) \dots \left( a\_{j\_{k\_{1}+l\_{1}}} \dots a\_{j\_{k\_{1}+1}j\_{k\_{1}}} a\_{j\_{k\_{1}}j} \right),$$

$$\boldsymbol{\tau} = \left( j\_{k\_{r}+l\_{r}} \dots j\_{k\_{r}+1} j\_{k\_{r}} \right) \dots \left( j\_{k\_{2}+l\_{2}} \dots j\_{k\_{2}+1} j\_{k\_{2}} \right) \left( j\_{k\_{1}+l\_{1}} \dots j\_{k\_{1}+1} j\_{k\_{1}} j \right),$$

with conditions, j <sup>k</sup><sup>2</sup> < j <sup>k</sup><sup>3</sup> < … < j kr and j kt < j ktþ<sup>s</sup> for <sup>t</sup> <sup>¼</sup> <sup>2</sup>, …, r and <sup>s</sup> <sup>¼</sup> <sup>1</sup>,…, lt.

Since rdet1<sup>A</sup> <sup>¼</sup> <sup>⋯</sup> <sup>¼</sup> rdetn<sup>A</sup> <sup>¼</sup> cdet1<sup>A</sup> <sup>¼</sup> <sup>⋯</sup> <sup>¼</sup> cdetn<sup>A</sup> <sup>∈</sup> <sup>R</sup> for Hermitian <sup>A</sup> <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>n</sup>, then we can define the determinant of a Hermitian matrix A by putting, detA≔rdet<sup>i</sup> A ¼ cdetiA, for all i ¼ 1, …, n. The determinant of a Hermitian matrix has properties similar to a usual determinant. They are completely explored in [17, 18] by its row and column determinants. In particular, within the framework of the theory of the column-row determinants, the determinantal representations of the inverse matrix over H by analogs of the classical adjoint matrix and Cramer's rule for quaternionic systems of linear equations have been derived. Further, we consider the determinantal representations of the Moore-Penrose inverse.

We shall use the following notations. Let α≔f g α1;…; α<sup>k</sup> ⊆f g 1;…; m and β≔ β1;…; β<sup>k</sup> � �⊆ f g <sup>1</sup>;…; <sup>n</sup> be subsets of the order 1 <sup>≤</sup> <sup>k</sup> <sup>≤</sup> minf g <sup>m</sup>; <sup>n</sup> . <sup>A</sup><sup>α</sup> <sup>β</sup> denotes the submatrix of A ∈ H<sup>n</sup>�<sup>m</sup> determined by the rows indexed by α and the columns indexed by β. Then, A<sup>α</sup> <sup>α</sup> denotes the principal submatrix determined by the rows and columns indexed by α. If A ∈ H<sup>n</sup>�<sup>n</sup> is Hermitian, then j j <sup>A</sup> <sup>α</sup> <sup>α</sup> is the corresponding principal minor of det A. For 1 ≤ k ≤ n, the collection of strictly increasing sequences of k integers chosen from 1f g ;…; n is denoted by Lk,n≔ α : α ¼ ð Þ α1; …; α<sup>k</sup> f g ; 1 ≤ α<sup>1</sup> ≤…≤ α<sup>k</sup> ≤ n . For fixed i∈ α and j∈ β, let Ir,mf gi ≔fα : α ∈ Lr,m; <sup>i</sup>∈αg, Jr,nf g<sup>j</sup> <sup>≔</sup> <sup>β</sup> : <sup>β</sup> <sup>∈</sup>Lr,n; <sup>j</sup>∈<sup>β</sup> � �.

Let a:<sup>j</sup> be the jth column and a<sup>i</sup>: be the ith row of A. Suppose A:<sup>j</sup>ð Þ b denotes the matrix obtained from A by replacing its jth column with the column b, then A<sup>i</sup>:ð Þ b denotes the matrix obtained from A by replacing its ith row with the row b. a<sup>∗</sup> :<sup>j</sup> and a<sup>∗</sup> <sup>i</sup>: denote the jth column and the ith row of A\* , respectively.

The following theorem gives determinantal representations of the Moore-Penrose inverse over the quaternion skew field H.

Theorem 2.1. [19] If <sup>A</sup> <sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>n</sup> <sup>r</sup> , then the Moore-Penrose inverse <sup>A</sup>† <sup>¼</sup> <sup>a</sup>† ij � � <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>m</sup> possesses the following determinantal representations:

$$a\_{ij}^{\dagger} = \frac{\sum\_{\beta \in I\_{r,u}\{i\}} \text{cdet}\_i \Big( (\mathbf{A}^\* \mathbf{A})\_{,i} \Big( \mathbf{a}\_{,j}^\* \Big) \Big)\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{r,u}} |\mathbf{A}^\* \mathbf{A}|\_{\beta}^{\beta}},\tag{2}$$

Theorem 2.3. [20] Let <sup>A</sup> <sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>n</sup> <sup>r</sup><sup>1</sup> and <sup>B</sup><sup>∈</sup> <sup>H</sup><sup>r</sup>�<sup>s</sup> <sup>r</sup><sup>2</sup> . Then, <sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup><sup>0</sup>

xij ¼

xij ¼

dB

dA

Lemma 3.1. [7] Let A<sup>1</sup> ∈ H<sup>m</sup>�<sup>n</sup>, B<sup>1</sup> ∈ H<sup>r</sup>�<sup>s</sup>

Then, the system (1) is consistent if and only if

<sup>1</sup> <sup>þ</sup> <sup>L</sup><sup>A</sup>1H†

<sup>þ</sup>L<sup>A</sup><sup>1</sup> <sup>Z</sup> � <sup>H</sup>†

:<sup>j</sup> <sup>¼</sup> <sup>X</sup>

<sup>i</sup>: <sup>¼</sup> <sup>X</sup>

2 4

2 4

α∈Ir2,rf gj

β∈Jr1,nf gi

P

P

P

P

<sup>β</sup>∈Jr1,nf g<sup>i</sup> cdet<sup>i</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>A</sup> :<sup>i</sup> <sup>d</sup><sup>B</sup>

β P

<sup>α</sup>∈Ir2,rf g<sup>j</sup> rdet<sup>j</sup> BB<sup>∗</sup> ð Þ<sup>j</sup>: <sup>d</sup><sup>A</sup>

β P

ð Þ ~c<sup>k</sup>: � �<sup>α</sup>

are the column vector and the row vector, respectively. ~c<sup>i</sup>: and ~c:<sup>j</sup> are the ith row and the jth column of

3. Determinantal representations of a partial solution to the system (1)

AiA† <sup>i</sup> CiB†

T A† 2XB† <sup>2</sup> � <sup>A</sup>†

In that case, the general solution of (1) can be expressed as the following,

2C2B†

<sup>2</sup> � <sup>A</sup>†

where Z and W are the arbitrary matrices over H with compatible dimensions.

� �B2B†

1C1B† 1

A2L<sup>T</sup> A†

HZB2B† 2 � � � <sup>L</sup><sup>A</sup>1H†

, C<sup>1</sup> ∈ H<sup>m</sup>�<sup>s</sup>

given and <sup>X</sup><sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> is to be determined. Put <sup>H</sup> <sup>¼</sup> <sup>A</sup>2L<sup>A</sup><sup>1</sup> , <sup>N</sup> <sup>¼</sup> <sup>R</sup><sup>B</sup>1B2, <sup>T</sup> <sup>¼</sup> <sup>R</sup>HA2, and <sup>F</sup> <sup>¼</sup> <sup>B</sup>2LN.

1C1B† 1

A2LTWNB†

<sup>2</sup> <sup>þ</sup> <sup>T</sup>†

α

β

3

3

<sup>β</sup>∈Jr1,n <sup>A</sup><sup>∗</sup> j j <sup>A</sup> <sup>β</sup>

<sup>β</sup>∈Jr1,n <sup>A</sup><sup>∗</sup> j j <sup>A</sup> <sup>β</sup>

rdet<sup>j</sup> BB<sup>∗</sup> ð Þ<sup>j</sup>:

cdet<sup>i</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>A</sup> :<sup>i</sup> <sup>C</sup><sup>~</sup> :<sup>l</sup> � � � � <sup>β</sup>

:j � � � � <sup>β</sup>

Cramer's Rules for the System of Two-Sided Matrix Equations and of Its Special Cases

i: � � � � <sup>α</sup>

<sup>α</sup><sup>∈</sup> Ir2,r BB<sup>∗</sup> j j<sup>α</sup>

<sup>5</sup><sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>1</sup>

<sup>5</sup><sup>∈</sup> <sup>H</sup><sup>1</sup>�<sup>r</sup>

<sup>α</sup><sup>∈</sup> Ir2,r BB<sup>∗</sup> j j<sup>α</sup>

β

α ,

α

α ,

, k ¼ 1, …, n,

, l ¼ 1, …, r,

, A<sup>2</sup> ∈ H<sup>k</sup>�<sup>n</sup>, B<sup>2</sup> ∈ H<sup>r</sup>�<sup>p</sup>

� �<sup>F</sup> <sup>¼</sup> <sup>0</sup>: (7)

T A† 2C2B†

<sup>2</sup> <sup>þ</sup> <sup>W</sup> � <sup>T</sup>†

<sup>i</sup> B<sup>i</sup> ¼ Ci, i ¼ 1, 2; (6)

<sup>2</sup> � <sup>A</sup>†

� �B2N†

1C1B† 1

TWNN† � � � <sup>R</sup><sup>B</sup><sup>1</sup> , (8)

, and C<sup>2</sup> ∈ H<sup>k</sup>�<sup>p</sup> be

R<sup>B</sup><sup>1</sup>

sentations,

or

where

<sup>C</sup><sup>~</sup> <sup>¼</sup> <sup>A</sup><sup>∗</sup>

<sup>X</sup> <sup>¼</sup> <sup>A</sup>†

1C1B†

CB<sup>∗</sup> . ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> has determinantal repre-

http://dx.doi.org/10.5772/intechopen.74105

7

or

$$a\_{ij}^{\dagger} = \frac{\sum\_{\alpha \in I\_{\prime,w}\{j\}} \text{rdet}\_{\circ} \Big( (\mathbf{A} \mathbf{A}^{\*})\_{j.} (\mathbf{a}\_{i.}^{\*}) \Big)\_{\alpha}^{\alpha}}{\sum\_{\alpha \in I\_{\prime,w}} |\mathbf{A} \mathbf{A}^{\*}|\_{\alpha}^{\alpha}}. \tag{3}$$

Remark 2.1. Note that for an arbitrary full-rank matrix, A ∈ H<sup>m</sup>�<sup>n</sup> <sup>r</sup> , a column-vector d:j, and a rowvector d<sup>i</sup>: with appropriate sizes, respectively, we put

cdet<sup>i</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>A</sup> :<sup>i</sup> <sup>d</sup>:<sup>j</sup> � � � � <sup>¼</sup> <sup>X</sup> β∈Jn,nf gi cdet<sup>i</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>A</sup> :<sup>i</sup> <sup>d</sup>:<sup>j</sup> � � � � <sup>β</sup> <sup>β</sup>, det <sup>A</sup><sup>∗</sup> ð Þ¼ <sup>A</sup> <sup>X</sup> β∈Jn,n <sup>A</sup><sup>∗</sup> j j <sup>A</sup> <sup>β</sup> <sup>β</sup> when r ¼ n, rdet<sup>j</sup> AA<sup>∗</sup> ð Þ<sup>j</sup>: ð Þ d<sup>i</sup>: � � <sup>¼</sup> <sup>X</sup> α∈ Im,mf gj rdet<sup>j</sup> AA<sup>∗</sup> ð Þ<sup>j</sup>: ð Þ d<sup>i</sup>: � �<sup>α</sup> α , det AA<sup>∗</sup> ð Þ¼ <sup>X</sup> α∈ Im,m AA<sup>∗</sup> j j<sup>α</sup> <sup>α</sup> when r ¼ m:

Furthermore, <sup>P</sup><sup>A</sup> <sup>¼</sup> <sup>A</sup>† <sup>A</sup>, <sup>Q</sup><sup>A</sup> <sup>¼</sup> AA† , <sup>L</sup><sup>A</sup> <sup>¼</sup> <sup>I</sup> � <sup>A</sup>† <sup>A</sup>, and <sup>R</sup>A≔<sup>I</sup> � AA† stand for some orthogonal projectors induced from A.

Theorem 2.2. [30] Let A ∈ H<sup>m</sup>�<sup>n</sup>, B∈ H<sup>r</sup>�<sup>s</sup> , and C ∈ H<sup>m</sup>�<sup>s</sup> be known and X∈ H<sup>n</sup>�<sup>r</sup> be unknown. Then, the matrix equation

$$\mathbf{A}\mathbf{X}\mathbf{B}=\mathbf{C}\tag{4}$$

is consistent if and only if AA† CBB† <sup>¼</sup> <sup>C</sup>. In this case, its general solution can be expressed as

$$\mathbf{X} = \mathbf{A}^{\dagger}\mathbf{C}\mathbf{B}^{\dagger} + \mathbf{L}\_{\mathrm{A}}\mathbf{V} + \mathbf{W}\mathbf{R}\_{\mathrm{B}}.\tag{5}$$

where V and W are arbitrary matrices over H with appropriate dimensions.

The partial solution, <sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>A</sup>† CB† , of (4) possesses the following determinantal representations. Cramer's Rules for the System of Two-Sided Matrix Equations and of Its Special Cases http://dx.doi.org/10.5772/intechopen.74105 7

Theorem 2.3. [20] Let <sup>A</sup> <sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>n</sup> <sup>r</sup><sup>1</sup> and <sup>B</sup><sup>∈</sup> <sup>H</sup><sup>r</sup>�<sup>s</sup> <sup>r</sup><sup>2</sup> . Then, <sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup><sup>0</sup> ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> has determinantal representations,

$$\mathbf{x}\_{ij} = \frac{\sum\_{\beta \in I\_{r\_1,n}\{i\}} \mathbf{cdet}\_i \left( (\mathbf{A}^\* \mathbf{A})\_{.j} \left( \mathbf{d}\_{.j}^B \right) \right)\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{r\_1,n}} |\mathbf{A}^\* \mathbf{A}|\_{\beta}^{\beta} \sum\_{\alpha \in I\_{r\_2,r}} |\mathbf{B} \mathbf{B}^\*|\_{\alpha}^{\alpha}}.$$

or

from A by replacing its ith row with the row b. a<sup>∗</sup>

of A\*

or

, respectively.

cdet<sup>i</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>A</sup> :<sup>i</sup> <sup>d</sup>:<sup>j</sup>

Furthermore, <sup>P</sup><sup>A</sup> <sup>¼</sup> <sup>A</sup>†

Then, the matrix equation

is consistent if and only if AA†

The partial solution, <sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>A</sup>†

rdet<sup>j</sup> AA<sup>∗</sup> ð Þ<sup>j</sup>:

the quaternion skew field H.

6 Matrix Theory-Applications and Theorems

following determinantal representations:

:<sup>j</sup> and a<sup>∗</sup>

The following theorem gives determinantal representations of the Moore-Penrose inverse over

<sup>β</sup>∈Jr,nf g<sup>i</sup> cdet<sup>i</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>A</sup> :<sup>i</sup> <sup>a</sup><sup>∗</sup>

<sup>α</sup><sup>∈</sup> Ir,mf g<sup>j</sup> rdet<sup>j</sup> AA<sup>∗</sup> ð Þ<sup>j</sup>: <sup>a</sup><sup>∗</sup>

Remark 2.1. Note that for an arbitrary full-rank matrix, A ∈ H<sup>m</sup>�<sup>n</sup> <sup>r</sup> , a column-vector d:j, and a row-

ð Þ d<sup>i</sup>: � �<sup>α</sup>

α

cdet<sup>i</sup> <sup>A</sup><sup>∗</sup> ð Þ <sup>A</sup> :<sup>i</sup> <sup>d</sup>:<sup>j</sup> � � � � <sup>β</sup>

, <sup>L</sup><sup>A</sup> <sup>¼</sup> <sup>I</sup> � <sup>A</sup>†

rdet<sup>j</sup> AA<sup>∗</sup> ð Þ<sup>j</sup>:

<sup>X</sup> <sup>¼</sup> <sup>A</sup>†

where V and W are arbitrary matrices over H with appropriate dimensions.

CB†

<sup>P</sup> <sup>α</sup> <sup>α</sup>∈Ir,m AA<sup>∗</sup> j j<sup>α</sup>

α

<sup>β</sup>∈Jr,n <sup>A</sup><sup>∗</sup> j j <sup>A</sup> <sup>β</sup>

P

:j � � � � <sup>β</sup>

i: � � � �<sup>α</sup>

<sup>β</sup>, det <sup>A</sup><sup>∗</sup> ð Þ¼ <sup>A</sup> <sup>X</sup>

, det AA<sup>∗</sup> ð Þ¼ <sup>X</sup>

CBB† <sup>¼</sup> <sup>C</sup>. In this case, its general solution can be expressed as

β∈Jn,n

α∈ Im,m

, and C ∈ H<sup>m</sup>�<sup>s</sup> be known and X∈ H<sup>n</sup>�<sup>r</sup> be unknown.

AXB ¼ C (4)

CB† <sup>þ</sup> <sup>L</sup>A<sup>V</sup> <sup>þ</sup> WRB, (5)

, of (4) possesses the following determinantal representations.

<sup>A</sup>, and <sup>R</sup>A≔<sup>I</sup> � AA† stand for some orthogo-

<sup>A</sup><sup>∗</sup> j j <sup>A</sup> <sup>β</sup>

AA<sup>∗</sup> j j<sup>α</sup>

β

β

Theorem 2.1. [19] If <sup>A</sup> <sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>n</sup> <sup>r</sup> , then the Moore-Penrose inverse <sup>A</sup>† <sup>¼</sup> <sup>a</sup>†

P

P

a† ij ¼

a† ij ¼

vector d<sup>i</sup>: with appropriate sizes, respectively, we put

β∈Jn,nf gi

<sup>A</sup>, <sup>Q</sup><sup>A</sup> <sup>¼</sup> AA†

<sup>¼</sup> <sup>X</sup> α∈ Im,mf gj

� � � � <sup>¼</sup> <sup>X</sup>

ð Þ d<sup>i</sup>: � �

nal projectors induced from A.

Theorem 2.2. [30] Let A ∈ H<sup>m</sup>�<sup>n</sup>, B∈ H<sup>r</sup>�<sup>s</sup>

<sup>i</sup>: denote the jth column and the ith row

ij � �

∈ H<sup>n</sup>�<sup>m</sup> possesses the

, (2)

: (3)

<sup>β</sup> when r ¼ n,

<sup>α</sup> when r ¼ m:

$$\mathbf{x}\_{i\circ} = \frac{\sum\_{\alpha \in I\_{r\_2,r}\{j\}} \mathbf{rdet}\_{\circ} \left( (\mathbf{BB}^\*)\_{\circ} \left( \mathbf{d}\_{i.}^A \right) \right)\_{\alpha}^{\alpha}}{\sum\_{\beta \in I\_{r\_1,r}} |\mathbf{A}^\* \mathbf{A}|\_{\beta}^{\beta} \sum\_{\alpha \in I\_{r\_2,r}} |\mathbf{BB}^\*|\_{\alpha}^{\alpha}} \,\prime$$

where

$$\mathbf{d}\_{\boldsymbol{j}}^{B} = \left[ \sum\_{\boldsymbol{\alpha} \in I\_{2r}, \{\boldsymbol{j}\}} \mathbf{r} \mathbf{det}\_{\boldsymbol{j}} \big( (\mathbf{B} \mathbf{B}^{\*})\_{\boldsymbol{j}}, (\tilde{\mathbf{c}}\_{k}) \big)\_{\boldsymbol{\alpha}}^{\boldsymbol{\alpha}} \right] \in \mathbb{H}^{n \times 1}, \ k = 1, \ldots, n,$$

$$\mathbf{d}\_{\boldsymbol{i}}^{A} = \left[ \sum\_{\boldsymbol{\beta} \in \tilde{I}\_{\boldsymbol{r}\_{l}, n} \{\boldsymbol{i}\}} \mathbf{c} \mathbf{det}\_{\boldsymbol{i}} \big( (\mathbf{A}^{\*} \mathbf{A})\_{\boldsymbol{i}} (\tilde{\mathbf{C}}\_{\boldsymbol{i}}) \big)\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \right] \in \mathbb{H}^{1 \times r}, \ l = 1, \ldots, r,$$

are the column vector and the row vector, respectively. ~c<sup>i</sup>: and ~c:<sup>j</sup> are the ith row and the jth column of <sup>C</sup><sup>~</sup> <sup>¼</sup> <sup>A</sup><sup>∗</sup> CB<sup>∗</sup> .
