4. Cramer's rules for special cases of (1)

In this section, we consider special cases of (1) when one or two equations are one-sided. Let in Eq.(1), the matrix B<sup>1</sup> is vanished. Then, we have the system

$$\begin{cases} \mathbf{A}\_1 \mathbf{X} = \mathbf{C}\_1, \\ \mathbf{A}\_2 \mathbf{X} \mathbf{B}\_2 = \mathbf{C}\_2. \end{cases} \tag{27}$$

The following lemma is extended to matrices with quaternion entries.

Lemma 4.1. [7] Let A<sup>1</sup> ∈ H<sup>m</sup>�<sup>n</sup>, C<sup>1</sup> ∈ H<sup>m</sup>�<sup>r</sup> , A<sup>2</sup> ∈ H<sup>k</sup>�<sup>n</sup>, B<sup>2</sup> ∈ H<sup>r</sup>�p, and C<sup>2</sup> ∈ H<sup>k</sup>�<sup>p</sup> be 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> . Then, the following statements are equivalent:

i. System (27) is consistent.

are the column vector and the row vector, respectively. c<sup>q</sup>:

<sup>β</sup>∈Jr7,nf g<sup>i</sup> cdet<sup>i</sup> <sup>T</sup><sup>∗</sup> ð Þ <sup>T</sup> :<sup>i</sup> <sup>a</sup>

<sup>f</sup> : are the <sup>q</sup>th column of <sup>T</sup>\*

, of the system (1), we have,

ij —(20), x<sup>05</sup>

4. Cramer's rules for special cases of (1)

Lemma 4.1. [7] Let A<sup>1</sup> ∈ H<sup>m</sup>�<sup>n</sup>, C<sup>1</sup> ∈ H<sup>m</sup>�<sup>r</sup>

Eq.(1), the matrix B<sup>1</sup> is vanished. Then, we have the system

P

<sup>2</sup> for all i ¼ 1, …, n and j ¼ 1,…, p.

<sup>β</sup><sup>∈</sup> Jr7,n <sup>T</sup><sup>∗</sup> j j <sup>T</sup> <sup>β</sup>

(vii) Using (3) for determinantal representations of and T† and (2) for N† in the seventh term of

ð Þ <sup>2</sup>;<sup>T</sup> :<sup>q</sup> � � � � <sup>β</sup>

> β P

Theorem 3.1. Let <sup>A</sup><sup>1</sup> <sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>n</sup> <sup>r</sup><sup>1</sup> , <sup>B</sup><sup>1</sup> <sup>∈</sup> <sup>H</sup><sup>r</sup>�<sup>s</sup> <sup>r</sup><sup>2</sup> , <sup>A</sup><sup>2</sup> <sup>∈</sup> <sup>H</sup><sup>k</sup>�<sup>n</sup> <sup>r</sup><sup>3</sup> , <sup>B</sup><sup>2</sup> <sup>∈</sup> <sup>H</sup><sup>r</sup>�<sup>p</sup> <sup>r</sup><sup>4</sup> , rank<sup>H</sup> <sup>¼</sup> rank <sup>A</sup>2L<sup>A</sup><sup>1</sup> ð Þ¼ <sup>r</sup>5, rankN ¼ ð Þ¼ R<sup>B</sup>1B<sup>2</sup> r6, and rankT ¼ ð Þ¼ RHA<sup>2</sup> r7. Then, for the partial solution (13),

ij has the determinantal representations (14) and (15), x<sup>02</sup>

In this section, we consider special cases of (1) when one or two equations are one-sided. Let in

A1X ¼ C1, A2XB<sup>2</sup> ¼ C2:

�

<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> . Then, the following statements are equivalent:

The following lemma is extended to matrices with quaternion entries.

ij —(23) and (24), and x<sup>07</sup>

x0 ij <sup>¼</sup> <sup>X</sup> δ x0δ

ij —(21), x<sup>06</sup>

β x<sup>01</sup> qf P

<sup>α</sup>∈Ir6,r NN<sup>∗</sup> j j<sup>α</sup>

α

<sup>A</sup><sup>2</sup> and the <sup>f</sup>th row of <sup>B</sup>2N<sup>∗</sup> <sup>¼</sup> <sup>B</sup>2B<sup>∗</sup>

<sup>l</sup>th column of <sup>C</sup><sup>2</sup> <sup>¼</sup> <sup>T</sup><sup>∗</sup>C2B<sup>∗</sup>

12 Matrix Theory-Applications and Theorems

ð Þ <sup>2</sup>;<sup>T</sup> :<sup>q</sup> and bð Þ <sup>2</sup>;<sup>N</sup>

∈ H<sup>n</sup>�<sup>r</sup>

where the term x<sup>01</sup>

ij —(18) and (19), x<sup>04</sup>

Hence, we prove the following theorem.

(13), we obtain

P<sup>n</sup> q¼1 P<sup>r</sup> f ¼1 P

x<sup>07</sup> ij ¼

where a

<sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup><sup>0</sup> ij � �

x<sup>03</sup>

tively.

ð Þ2 and c:<sup>l</sup>

ð Þ2 are the qth row and the

f : � � � � <sup>α</sup>

α

<sup>2</sup>R<sup>B</sup><sup>1</sup> , respec-

ij —(16) and (17),

(27)

,

(25)

<sup>α</sup>∈Ir6,rf g<sup>j</sup> rdet<sup>j</sup> NN<sup>∗</sup> ð Þ<sup>j</sup>: <sup>b</sup>ð Þ <sup>2</sup>;<sup>N</sup>

ij , (26)

ij —(25).

, A<sup>2</sup> ∈ H<sup>k</sup>�<sup>n</sup>, B<sup>2</sup> ∈ H<sup>r</sup>�p, and C<sup>2</sup> ∈ H<sup>k</sup>�<sup>p</sup> be given and

$$\text{iii.} \quad \mathbf{R}\_{A\_1}\mathbf{C}\_1 = \mathbf{0},\\\mathbf{R}\_H(\mathbf{C}\_2 - \mathbf{A}\_2\mathbf{A}\_1^\dagger\mathbf{C}\_1\mathbf{B}\_2) = \mathbf{0},\\\mathbf{C}\_2\mathbf{L}\_{B\_2} = \mathbf{0}.$$

$$\mathbf{iii}\mathbf{i} \mathbf{i} \quad \text{rank}[\mathbf{A}\_1 \ \mathbf{C}\_1] = \text{rank}[\mathbf{A}\_1], \text{rank}\begin{bmatrix} \mathbf{C}\_2 \\ \mathbf{B}\_2 \end{bmatrix} = \text{rank}[\mathbf{B}\_2], \text{rank}\begin{bmatrix} \mathbf{A}\_1 & \mathbf{C}\_1 \mathbf{B}\_2 \\ \mathbf{A}\_2 & \mathbf{C}\_2 \end{bmatrix} = \text{rank}\begin{bmatrix} \mathbf{A}\_1 \\ \mathbf{A}\_2 \end{bmatrix}.$$

In this case, the general solution of (27) can be expressed as

$$\mathbf{X} = \mathbf{A}\_1^\dagger \mathbf{C}\_1 + \mathbf{L}\_{A\_1} \mathbf{H}^\dagger \left(\mathbf{C}\_2 - \mathbf{A}\_2 \mathbf{A}\_1^\dagger \mathbf{C}\_1 \mathbf{B}\_2\right) \mathbf{B}\_2^\dagger + \mathbf{L}\_{A\_1} \mathbf{L}\_H \mathbf{Z}\_1 + \mathbf{L}\_{A\_1} \mathbf{W}\_1 \mathbf{R}\_{\overline{\mathbf{B}}\_2} \tag{28}$$

where Z<sup>1</sup> and W<sup>1</sup> are the arbitrary matrices over H with appropriate sizes.

Since by (9), <sup>L</sup><sup>A</sup>1H† <sup>¼</sup> <sup>L</sup><sup>A</sup><sup>1</sup> <sup>A</sup>2L<sup>A</sup><sup>1</sup> ð Þ† <sup>¼</sup> <sup>A</sup>2L<sup>A</sup><sup>1</sup> ð Þ† <sup>¼</sup> <sup>H</sup>† , then we have some simplification of (28),

$$\mathbf{X} = \mathbf{A}\_1^\dagger \mathbf{C}\_1 + \mathbf{H}^\dagger \mathbf{C}\_2 \mathbf{B}\_2^\dagger - \mathbf{H}^\dagger \mathbf{A}\_2 \mathbf{A}\_1^\dagger \mathbf{C}\_1 \mathbf{B}\_2 \mathbf{B}\_2^\dagger + \mathbf{L}\_{A\_1} \mathbf{L}\_H \mathbf{Z}\_1 + \mathbf{L}\_{A\_1} \mathbf{W}\_1 \mathbf{R}\_{\partial\_2}.$$

By putting Z1=W1=0, there is the following partial solution of (27),

$$\mathbf{X}\_{0} = \mathbf{A}\_{1}^{\dagger}\mathbf{C}\_{1} + \mathbf{H}^{\dagger}\mathbf{C}\_{2}\mathbf{B}\_{2}^{\dagger} - \mathbf{H}^{\dagger}\mathbf{A}\_{2}\mathbf{A}\_{1}^{\dagger}\mathbf{C}\_{1}\mathbf{B}\_{2}\mathbf{B}\_{2}^{\dagger}.\tag{29}$$

Theorem 4.1. Let A<sup>1</sup> ¼ a ð Þ1 ij � �<sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>n</sup> <sup>r</sup><sup>1</sup> , <sup>A</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup> ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>k</sup>�<sup>n</sup> <sup>r</sup><sup>2</sup> , <sup>B</sup><sup>2</sup> <sup>¼</sup> <sup>b</sup> ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>r</sup>�<sup>p</sup> <sup>r</sup><sup>3</sup> , C<sup>1</sup> ¼ c ð Þ1 ij � �<sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>r</sup> , and C<sup>2</sup> ¼ c ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>k</sup>�<sup>p</sup> , and there exist A† <sup>1</sup> ¼ a ð Þ1 ,† ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>m</sup>, B† <sup>2</sup> ¼ b ð Þ2 ,† ij � � <sup>∈</sup> <sup>H</sup><sup>p</sup>�<sup>r</sup> , and <sup>H</sup>† <sup>¼</sup> <sup>h</sup>† ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>k</sup> . Let rankH ¼ min rankA2;rankL<sup>A</sup><sup>1</sup> f g ¼ r4. Denote A∗ <sup>1</sup>C1≕C<sup>b</sup> <sup>1</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup> ð Þ1 ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> , H<sup>∗</sup> C2B<sup>∗</sup> <sup>2</sup>≕Cb<sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup> ð Þ2 ij � � <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> , H<sup>∗</sup>A2A<sup>∗</sup> <sup>1</sup>≕A<sup>b</sup> <sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>a</sup> ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�m, and <sup>C</sup>1Q<sup>B</sup>2≕Q<sup>b</sup> <sup>¼</sup> <sup>b</sup>qij � �<sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>p</sup> . Then, the partial solution (29), <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> , possesses the following determinantal representations,

$$\begin{split} \mathbf{x}\_{ij}^{0} &= \frac{\sum\_{\beta \in I\_{\tau\_{1},n} \{i\}} \mathsf{cdet}\_{i} \Big( \left( \mathbf{A}\_{1}^{\*} \mathbf{A}\_{1} \right)\_{:,i} \Big( \widehat{\mathbf{c}}\_{\cdot,j}^{(1)} \Big) \Big)\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{\tau\_{1},n} \mid \mathbf{A}\_{1}^{\*} \mathbf{A}\_{1} \big|\_{\beta}^{\beta}} + \frac{d\_{ij}^{(\lambda)}}{\sum\_{\beta \in I\_{\tau\_{4},n} \mid \mathbf{I}} \Big| \mathbf{H}^{\*} \mathbf{H} \big|\_{\beta}^{\beta} \sum\_{\alpha \in I\_{\tau\_{3},n}} \Big| \mathbf{B}\_{2} \mathbf{B}\_{2}^{\*} \big|\_{\alpha}^{\alpha}} \\ &- \frac{\sum\_{l=1}^{m} g\_{il}^{(\mu)} \sum\_{\alpha \in I\_{\tau\_{3},n} \mid \mathbf{j}} \mathsf{cdet}\_{l} \Big( \left( \mathbf{B}\_{2} \mathbf{B}\_{2}^{\*} \right)\_{\boldsymbol{j}} \big( \widehat{\mathbf{q}}\_{l} \big) \Big)\_{\alpha}^{\alpha}}{\sum\_{\beta \in I\_{\tau\_{4},n} \mid \mathbf{i}} |\mathbf{H}^{\*} \mathbf{H}|\_{\beta}^{\beta} \sum\_{\alpha \in I\_{\tau\_{1},n}} \Big| \mathbf{A}\_{1} \mathbf{A}\_{1}^{\*} \big|\_{\alpha}^{\alpha} \sum\_{\alpha \in I\_{\tau\_{3},n}} \Big| \mathbf{B}\_{2} \mathbf{B}\_{2}^{\*} \big|\_{\alpha}^{\alpha}} \end{split}$$

for all λ ¼ 1, 2 and μ ¼ 1, 2. Here

$$d\_{ij}^{(1)} \coloneqq \sum\_{a \in I\_{2,r}} \mathbf{rdet}\_{\{j\}} \Big( \left( \mathbf{B}\_2 \mathbf{B}\_2^\* \right)\_{\not{j}} \Big( \mathbf{v}\_{i.}^{(1)} \Big) \Big)\_{a'}^{a} \\ g\_{il}^{(1)} \coloneqq \sum\_{a \in I\_{1,n}} \mathbf{rdet}\_{\!\!\!/} \Big( \left( \mathbf{A}\_1 \mathbf{A}\_1^\* \right)\_{\!\!\!/} \Big( \mathbf{u}\_{i.}^{(1)} \Big) \Big)\_{a'}^{a}$$

and the row-vectors vð Þ<sup>1</sup> <sup>i</sup>: ¼ v ð Þ1 <sup>i</sup><sup>1</sup> ;…; v ð Þ1 ir � � and <sup>u</sup>ð Þ<sup>1</sup> <sup>i</sup>: <sup>¼</sup> <sup>u</sup>ð Þ<sup>1</sup> <sup>i</sup><sup>1</sup> ;…; <sup>u</sup>ð Þ<sup>1</sup> im � � such that

$$w\_{it}^{(1)} \coloneqq \sum\_{\boldsymbol{\beta} \in \mathcal{I}\_{\boldsymbol{\tau}\_{\boldsymbol{\theta}^\*}, \boldsymbol{\epsilon}}\{i\}} \mathsf{cdet}\_{i}\Big((\mathbf{H}^\*\mathbf{H})\_{\boldsymbol{\cdot}}\Big(\widehat{\mathbf{c}}\_{\boldsymbol{\cdot}}^{(2)}\Big)\Big)\_{\boldsymbol{\beta}'}^{\boldsymbol{\beta}} \,\boldsymbol{u}\_{iz}^{(1)} \coloneqq \sum\_{\boldsymbol{\beta} \in \mathcal{I}\_{\boldsymbol{\tau}\_{\boldsymbol{\theta}^\*}, \boldsymbol{\epsilon}}\{i\}} \mathsf{cdet}\_{i}\Big((\mathbf{H}^\*\mathbf{H})\_{\boldsymbol{\cdot}}\Big(\widehat{\mathbf{a}}\_{\boldsymbol{z}}^{(2)}\Big)\Big)\_{\boldsymbol{\beta}'}^{\boldsymbol{\beta}}$$

In another case,

$$d\_{ij}^{(2)} \coloneqq \sum\_{\boldsymbol{\beta} \in \mathcal{I}\_{\mathbf{r}\_{\mathbf{d}}, \boldsymbol{\pi}}\{i\}} \mathsf{cdet}\_{i} \Big( (\mathbf{H}^{\*}\mathbf{H})\_{:i} \Big( \mathbf{v}\_{:j}^{(2)} \Big) \Big)\_{\boldsymbol{\beta}'}^{\boldsymbol{\beta}} g\_{il}^{(2)} \coloneqq \sum\_{\boldsymbol{\beta} \in \mathcal{I}\_{\mathbf{r}\_{\mathbf{d}}, \boldsymbol{\pi}}\{i\}} \mathsf{cdet}\_{i} \Big( (\mathbf{H}^{\*}\mathbf{H})\_{:i} \Big( \mathbf{u}\_{.l}^{(2)} \Big) \Big)\_{\boldsymbol{\beta}'}^{\boldsymbol{\beta}}$$

and the column-vectors vð Þ<sup>2</sup> :<sup>j</sup> ¼ v ð Þ2 <sup>1</sup><sup>j</sup> ;…; v ð Þ2 nj � � and <sup>u</sup>ð Þ<sup>2</sup> :<sup>l</sup> <sup>¼</sup> <sup>u</sup>ð Þ<sup>2</sup> <sup>1</sup><sup>l</sup> ;…; <sup>u</sup>ð Þ<sup>2</sup> nl � � such that

$$\boldsymbol{\sigma}\_{q\circ}^{(2)} \coloneqq \sum\_{\boldsymbol{\alpha} \in I\_{r\_{\mathcal{I}},r}\{\boldsymbol{\beta}\}} \mathbf{rdet}\_{\boldsymbol{\beta}} \Big( \left( \mathbf{B}\_{2} \mathbf{B}\_{2}^{\*} \right)\_{\boldsymbol{\beta}} \Big( \hat{\mathbf{c}}\_{q\cdot}^{(2)} \Big) \Big)\_{\boldsymbol{\alpha}'}^{a} \,\boldsymbol{u}\_{q\prime}^{(2)} \coloneqq \sum\_{\boldsymbol{\alpha} \in I\_{r\_{\mathcal{I}},n}\{\boldsymbol{l}\}} \mathbf{rdet}\_{\boldsymbol{l}} \Big( \left( \mathbf{A}\_{1} \mathbf{A}\_{1}^{\*} \right)\_{\boldsymbol{l}} \Big( \boldsymbol{a}\_{q\cdot}^{(2)} \Big) \Big)\_{\boldsymbol{a}}^{a}$$

Proof. The proof is similar to the proof of Theorem 3.1.

Let in Eq.(1), the matrix A<sup>1</sup> is vanished. Then, we have the system,

$$\begin{cases} \mathbf{X} \mathbf{B}\_1 = \mathbf{C}\_1, \\ \mathbf{A}\_2 \mathbf{X} \mathbf{B}\_2 = \mathbf{C}\_2. \end{cases} \tag{30}$$

:

Theorem 4.2. Let B<sup>1</sup> ¼ b

the partial solution (32), <sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup><sup>0</sup>

P

�

for all λ ¼ 1, 2 and μ ¼ 1, 2. Here

d ð Þ1 ij <sup>≔</sup> <sup>X</sup> α∈Ir3,rf gj

and the row-vectors φð Þ<sup>1</sup>

φð Þ<sup>1</sup>

In another case,

d ð Þ2 ij <sup>≔</sup> <sup>X</sup> β∈Jr2,nf gi

and the column-vectors φð Þ<sup>2</sup>

φð Þ<sup>2</sup>

qj <sup>¼</sup> <sup>X</sup>

α ∈Ir4,rf gj

are vanished. Then, we have the system

it <sup>¼</sup> <sup>X</sup>

β∈ Jr2,nf gi

P

P<sup>s</sup> z¼1 P

<sup>β</sup><sup>∈</sup> Jr2,n <sup>A</sup><sup>∗</sup>

<sup>i</sup>: <sup>¼</sup> <sup>φ</sup>ð Þ<sup>1</sup>

cdet<sup>i</sup> A<sup>∗</sup>

cdet<sup>i</sup> A<sup>∗</sup>

:<sup>j</sup> <sup>¼</sup> <sup>φ</sup>ð Þ<sup>2</sup>

ij � �<sup>∈</sup> <sup>H</sup><sup>p</sup>�<sup>r</sup>

x0 ij ¼

<sup>2</sup>C2N∗≕C<sup>~</sup> <sup>2</sup> <sup>¼</sup> <sup>~</sup><sup>c</sup>

, and C<sup>2</sup> ¼ c

ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup>

c ð Þ1 ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>s</sup>

A∗

<sup>N</sup>† <sup>¼</sup> <sup>n</sup>†

ð Þ1

, B<sup>∗</sup> 1B2N<sup>∗</sup>

<sup>α</sup>∈Ir1,rf g<sup>j</sup> rdet<sup>j</sup> <sup>B</sup>1B<sup>∗</sup>

ij � � <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup>

� � � � α α

<sup>2</sup>A<sup>2</sup> � � � � β β P

rdet<sup>j</sup> NN<sup>∗</sup> ð Þ<sup>j</sup>: <sup>φ</sup>ð Þ<sup>1</sup>

<sup>i</sup><sup>1</sup> ;…;φð Þ<sup>1</sup> ir � � and <sup>ψ</sup>ð Þ<sup>1</sup>

> <sup>2</sup>A<sup>2</sup> � �

<sup>2</sup>A<sup>2</sup> � �

rdet<sup>j</sup> NN<sup>∗</sup> ð Þ<sup>j</sup>: <sup>c</sup>

The following lemma is extended to matrices with quaternion entries.

:<sup>i</sup> c ð Þ2 :t � � � � <sup>β</sup>

:<sup>i</sup> <sup>φ</sup>ð Þ<sup>2</sup> :j � � � � <sup>β</sup>

> ð Þ2 q: � � � � <sup>α</sup>

> > �

<sup>1</sup><sup>j</sup> ;…;φð Þ<sup>2</sup> nj � � and <sup>ψ</sup>ð Þ<sup>2</sup>

1 � �

1

<sup>β</sup><sup>∈</sup> Jr2,nf g<sup>i</sup> cdet<sup>i</sup> <sup>A</sup><sup>∗</sup>

<sup>P</sup> <sup>α</sup> <sup>α</sup><sup>∈</sup> Ir1,r <sup>B</sup>1B<sup>∗</sup>

<sup>j</sup>: ~c ð Þ1 i: � � � � <sup>α</sup>

<sup>β</sup><sup>∈</sup> Jr1,s <sup>B</sup><sup>∗</sup>

i: � � � � <sup>α</sup>

ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>k</sup>�<sup>p</sup>

ij � � <sup>∈</sup> <sup>H</sup><sup>r</sup>�<sup>s</sup> <sup>r</sup><sup>1</sup> , <sup>A</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup>

, and there exist B†

. Let rank<sup>N</sup> <sup>¼</sup> min rankB2;rankR<sup>B</sup><sup>1</sup> <sup>f</sup> g ¼ <sup>r</sup>4. Denote <sup>C</sup>1B<sup>∗</sup>

ð Þ2 ij � � <sup>∈</sup> <sup>H</sup><sup>s</sup>�<sup>r</sup>

þ

<sup>2</sup>A<sup>2</sup> � �

<sup>1</sup>B<sup>1</sup> � � � � β β P

> α , gð Þ<sup>1</sup>

β , ψð Þ<sup>1</sup>

β , gð Þ<sup>2</sup>

α , ψð Þ<sup>2</sup>

Now, suppose that the both equations of (1) are one-sided. Let in Eq.(1), the matrices B<sup>1</sup> and A<sup>2</sup>

A1X ¼ C1, XB<sup>2</sup> ¼ C2:

P

:<sup>i</sup> p~:<sup>z</sup> � � � �<sup>β</sup>

> il <sup>≔</sup> <sup>X</sup> α∈ Ir4,rf gj

zv <sup>¼</sup> <sup>X</sup>

zj <sup>≔</sup> <sup>X</sup> β∈ Jr1,nf gz

:<sup>j</sup> <sup>¼</sup> <sup>ψ</sup>ð Þ<sup>2</sup>

uj <sup>≔</sup> <sup>X</sup> α∈Ir4,rf gj

<sup>i</sup>: <sup>¼</sup> <sup>ψ</sup>ð Þ<sup>1</sup>

<sup>β</sup>∈Jr2,n <sup>A</sup><sup>∗</sup>

β g ð Þ<sup>μ</sup> zj

<sup>α</sup><sup>∈</sup> Ir4,r NN<sup>∗</sup> j j<sup>α</sup>

<sup>z</sup><sup>1</sup> ; …;ψð Þ<sup>1</sup> zr � � such that

β∈Jr1,nf gz

<sup>≕</sup>B~<sup>2</sup> <sup>¼</sup> <sup>~</sup><sup>b</sup>

ð Þ2

<sup>1</sup> ¼ b

ij � � <sup>∈</sup> <sup>H</sup><sup>k</sup>�<sup>n</sup> <sup>r</sup><sup>2</sup> , <sup>B</sup><sup>2</sup> <sup>¼</sup> <sup>b</sup>

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

ð Þ1 , † ij � �<sup>∈</sup> <sup>H</sup><sup>s</sup>�<sup>r</sup>

, possesses the following determinantal representations,

d ð Þ λ ij

<sup>2</sup>A<sup>2</sup> � � � � β β P

α

rdet<sup>j</sup> NN<sup>∗</sup> ð Þ<sup>j</sup>: <sup>ψ</sup><sup>z</sup>: � � � �<sup>α</sup>

cdet<sup>z</sup> B<sup>∗</sup>

cdet<sup>z</sup> B<sup>∗</sup>

<sup>1</sup><sup>j</sup> ;…; <sup>ψ</sup>ð Þ<sup>2</sup> sj � � such that

<sup>1</sup>B<sup>1</sup> � �

<sup>1</sup>B<sup>1</sup> � �

rdet<sup>j</sup> NN<sup>∗</sup> ð Þ<sup>j</sup>: <sup>b</sup>ð Þ<sup>2</sup>

:<sup>i</sup> <sup>b</sup>ð Þ<sup>2</sup> :v � � � � <sup>β</sup>

:<sup>z</sup> <sup>ψ</sup>ð Þ<sup>2</sup> :j � � � � <sup>β</sup>

u: � � � � <sup>α</sup> β :

β ,

α :

(33)

ð Þ2

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

<sup>1</sup>≕C<sup>~</sup> <sup>1</sup> <sup>¼</sup> <sup>~</sup><sup>c</sup>

<sup>α</sup><sup>∈</sup> Ir4,r NN<sup>∗</sup> j j<sup>α</sup>

α

α ,

, A† <sup>2</sup> ¼ a

, and <sup>P</sup><sup>A</sup>2C1≕P<sup>~</sup> <sup>¼</sup> <sup>p</sup>~ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>s</sup>

ij � �<sup>∈</sup> <sup>H</sup><sup>r</sup>�<sup>p</sup> <sup>r</sup><sup>3</sup> , <sup>C</sup><sup>1</sup> <sup>¼</sup>

ð Þ2 ,† ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>k</sup>

> ð Þ1 ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup>

,

15

,

. Then,

The following lemma is extended to matrices with quaternion entries as well.

Lemma 4.2. [7] Let B<sup>1</sup> ∈ H<sup>r</sup>�<sup>s</sup> , C<sup>1</sup> ∈ H<sup>n</sup>�<sup>s</sup> , A<sup>2</sup> ∈ H<sup>k</sup>�<sup>n</sup>, B<sup>2</sup> ∈ H<sup>r</sup>�<sup>p</sup> , and C<sup>2</sup> ∈ H<sup>k</sup>�<sup>p</sup> be given and <sup>X</sup><sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> is to be determined. Put <sup>N</sup> <sup>¼</sup> <sup>R</sup><sup>B</sup>1B2. Then, the following statements are equivalent:

i. System (30) is consistent.

$$\text{iii}\qquad \mathbf{R}\_{A\_2}\mathbf{C}\_2 = \mathbf{0},\\ \left(\mathbf{C}\_2 - \mathbf{A}\_2\mathbf{C}\_1\mathbf{B}\_1^\dagger\mathbf{B}\_2\right)\mathbf{L}\_N = \mathbf{0},\\ \mathbf{C}\_2\mathbf{L}\_{B\_2} = \mathbf{0}.$$

$$\mathbf{iii}\mathbf{i} \quad \text{rank}[\mathbf{A}\_2 \quad \mathbf{C}\_2] = \text{rank}[\mathbf{A}\_2], \text{ rank}\begin{bmatrix} \mathbf{C}\_1\\ \mathbf{B}\_1 \end{bmatrix} = \text{rank}[\mathbf{B}\_1], \text{ rank}\begin{bmatrix} \mathbf{C}\_2 & \mathbf{A}\_2 \mathbf{C}\_1\\ \mathbf{B}\_2 & \mathbf{B}\_1 \end{bmatrix} = \text{rank}[\mathbf{B}\_2 \quad \mathbf{B}\_1].$$

In this case, the general solution of (30) can be expressed as

$$\mathbf{X} = \mathbf{C}\_1 \mathbf{B}\_1^\dagger + \mathbf{A}\_2^\dagger (\mathbf{C}\_2 - \mathbf{A}\_2 \mathbf{C}\_1 \mathbf{B}\_1^\dagger \mathbf{B}\_2) \mathbf{N}^\dagger \mathbf{R}\_{\overline{\theta}\_1} + \mathbf{L}\_{A\_2} \mathbf{W}\_2 \mathbf{R}\_{\overline{\theta}\_1} + \mathbf{Z}\_2 \mathbf{R}\_N \mathbf{R}\_{\overline{\theta}\_1} \tag{31}$$

where Z<sup>2</sup> and W<sup>2</sup> are the arbitrary matrices over H with appropriate sizes.

Since by (10), N† R<sup>B</sup><sup>1</sup> ¼ ð Þ R<sup>B</sup>1B<sup>2</sup> † <sup>R</sup><sup>B</sup><sup>1</sup> <sup>¼</sup> <sup>N</sup>† , then some simplification of (31) can be derived,

$$\mathbf{X} = \mathbf{C}\_1 \mathbf{B}\_1^\dagger + \mathbf{A}\_2^\dagger \mathbf{C}\_2 \mathbf{N}^\dagger - \mathbf{A}\_2 \mathbf{C}\_1 \mathbf{B}\_1^\dagger \mathbf{B}\_2 \mathbf{N}^\dagger + \mathbf{L}\_{A\_2} \mathbf{W}\_2 \mathbf{R}\_{B\_1} + \mathbf{Z}\_2 \mathbf{R}\_N \mathbf{R}\_{B\_1}.$$

By putting Z2=W2=0, there is the following partial solution of (30),

$$\mathbf{X}\_{0} = \mathbf{C}\_{1}\mathbf{B}\_{1}^{\dagger} + \mathbf{A}\_{2}^{\dagger}\mathbf{C}\_{2}\mathbf{N}^{\dagger} - \mathbf{A}\_{2}^{\dagger}\mathbf{A}\_{2}\mathbf{C}\_{1}\mathbf{B}\_{1}^{\dagger}\mathbf{B}\_{2}\mathbf{N}^{\dagger}.\tag{32}$$

The following theorem on determinantal representations of (29) can be proven similar to the proof of Theorem 3.1 as well.

Cramer's Rules for the System of Two-Sided Matrix Equations and of Its Special Cases http://dx.doi.org/10.5772/intechopen.74105 15

Theorem 4.2. Let B<sup>1</sup> ¼ b ð Þ1 ij � � <sup>∈</sup> <sup>H</sup><sup>r</sup>�<sup>s</sup> <sup>r</sup><sup>1</sup> , <sup>A</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup> ð Þ2 ij � � <sup>∈</sup> <sup>H</sup><sup>k</sup>�<sup>n</sup> <sup>r</sup><sup>2</sup> , <sup>B</sup><sup>2</sup> <sup>¼</sup> <sup>b</sup> ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>r</sup>�<sup>p</sup> <sup>r</sup><sup>3</sup> , <sup>C</sup><sup>1</sup> <sup>¼</sup> c ð Þ1 ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>s</sup> , and C<sup>2</sup> ¼ c ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>k</sup>�<sup>p</sup> , and there exist B† <sup>1</sup> ¼ b ð Þ1 , † ij � �<sup>∈</sup> <sup>H</sup><sup>s</sup>�<sup>r</sup> , A† <sup>2</sup> ¼ a ð Þ2 ,† ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>k</sup> , <sup>N</sup>† <sup>¼</sup> <sup>n</sup>† ij � �<sup>∈</sup> <sup>H</sup><sup>p</sup>�<sup>r</sup> . Let rank<sup>N</sup> <sup>¼</sup> min rankB2;rankR<sup>B</sup><sup>1</sup> <sup>f</sup> g ¼ <sup>r</sup>4. Denote <sup>C</sup>1B<sup>∗</sup> <sup>1</sup>≕C<sup>~</sup> <sup>1</sup> <sup>¼</sup> <sup>~</sup><sup>c</sup> ð Þ1 ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> , A∗ <sup>2</sup>C2N∗≕C<sup>~</sup> <sup>2</sup> <sup>¼</sup> <sup>~</sup><sup>c</sup> ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> , B<sup>∗</sup> 1B2N<sup>∗</sup> <sup>≕</sup>B~<sup>2</sup> <sup>¼</sup> <sup>~</sup><sup>b</sup> ð Þ2 ij � � <sup>∈</sup> <sup>H</sup><sup>s</sup>�<sup>r</sup> , and <sup>P</sup><sup>A</sup>2C1≕P<sup>~</sup> <sup>¼</sup> <sup>p</sup>~ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>s</sup> . Then, the partial solution (32), <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> , possesses the following determinantal representations,

x0 ij ¼ P <sup>α</sup>∈Ir1,rf g<sup>j</sup> rdet<sup>j</sup> <sup>B</sup>1B<sup>∗</sup> 1 � � <sup>j</sup>: ~c ð Þ1 i: � � � � <sup>α</sup> <sup>P</sup> <sup>α</sup> <sup>α</sup><sup>∈</sup> Ir1,r <sup>B</sup>1B<sup>∗</sup> 1 � � � � α α þ d ð Þ λ ij P <sup>β</sup>∈Jr2,n <sup>A</sup><sup>∗</sup> <sup>2</sup>A<sup>2</sup> � � � � β β P <sup>α</sup><sup>∈</sup> Ir4,r NN<sup>∗</sup> j j<sup>α</sup> α � P<sup>s</sup> z¼1 P <sup>β</sup><sup>∈</sup> Jr2,nf g<sup>i</sup> cdet<sup>i</sup> <sup>A</sup><sup>∗</sup> <sup>2</sup>A<sup>2</sup> � � :<sup>i</sup> p~:<sup>z</sup> � � � �<sup>β</sup> β g ð Þ<sup>μ</sup> zj P <sup>β</sup><sup>∈</sup> Jr2,n <sup>A</sup><sup>∗</sup> <sup>2</sup>A<sup>2</sup> � � � � β β P <sup>β</sup><sup>∈</sup> Jr1,s <sup>B</sup><sup>∗</sup> <sup>1</sup>B<sup>1</sup> � � � � β β P <sup>α</sup><sup>∈</sup> Ir4,r NN<sup>∗</sup> j j<sup>α</sup> α

for all λ ¼ 1, 2 and μ ¼ 1, 2. Here

$$d\_{\vec{\boldsymbol{\eta}}}^{(1)} \coloneqq \sum\_{\boldsymbol{\alpha} \in \boldsymbol{l}\_{\triangleright^\*} \{\boldsymbol{\beta}\}} \mathbf{rdet}\_{\boldsymbol{\beta}} \Big( (\mathbf{NN}^\*)\_{\boldsymbol{\beta}.} \Big( \boldsymbol{\varrho}\_{\boldsymbol{\iota}.}^{(1)} \Big) \Big)\_{\boldsymbol{\alpha}'}^{\boldsymbol{\alpha}} \cdot g\_{\boldsymbol{\alpha}}^{(1)} \coloneqq \sum\_{\boldsymbol{\alpha} \in \boldsymbol{l}\_{\boldsymbol{l}\_{\mathtt{\mathsf{s}}}} \{\boldsymbol{\beta}\}} \mathbf{rdet}\_{\boldsymbol{\beta}} \Big( (\mathbf{NN}^\*)\_{\boldsymbol{\beta}.} \Big( \boldsymbol{\psi}\_{\boldsymbol{z}} \Big) \Big)\_{\boldsymbol{\alpha}'}^{\boldsymbol{\alpha}}$$

and the row-vectors φð Þ<sup>1</sup> <sup>i</sup>: <sup>¼</sup> <sup>φ</sup>ð Þ<sup>1</sup> <sup>i</sup><sup>1</sup> ;…;φð Þ<sup>1</sup> ir � � and <sup>ψ</sup>ð Þ<sup>1</sup> <sup>i</sup>: <sup>¼</sup> <sup>ψ</sup>ð Þ<sup>1</sup> <sup>z</sup><sup>1</sup> ; …;ψð Þ<sup>1</sup> zr � � such that

$$\boldsymbol{\varphi}\_{\boldsymbol{\upbeta}}^{(1)} = \sum\_{\boldsymbol{\upbeta} \in \mathcal{I}\_{\boldsymbol{\upbeta},n} \{i\}} \mathbf{c} \mathbf{det}\_{i} \big( \left( \mathbf{A}\_{2}^{\*} \mathbf{A}\_{2} \right)\_{.i} \big( \mathbf{c}\_{.t}^{(2)} \big) \big)\_{\boldsymbol{\upbeta}'}^{\boldsymbol{\upbeta}} \,\psi\_{\boldsymbol{z}\boldsymbol{v}}^{(1)} = \sum\_{\boldsymbol{\upbeta} \in \mathcal{I}\_{\boldsymbol{\upbeta},n} \{\boldsymbol{z}\}} \mathbf{c} \mathbf{det}\_{i} \big( \left( \mathbf{B}\_{1}^{\*} \mathbf{B}\_{1} \right)\_{.i} \big( \mathbf{b}\_{.\boldsymbol{\upbeta}}^{(2)} \big) \big)\_{\boldsymbol{\upbeta}}^{\boldsymbol{\upbeta}} \,\psi\_{\boldsymbol{z}\boldsymbol{v}}^{(2)}$$

In another case,

v ð Þ1 it <sup>≔</sup> <sup>X</sup> β∈Jr4,nf gi

14 Matrix Theory-Applications and Theorems

d ð Þ2 ij <sup>≔</sup> <sup>X</sup> β∈ Jr4,nf gi

v ð Þ2 qj <sup>≔</sup> <sup>X</sup> α∈ Ir3,rf gj

and the column-vectors vð Þ<sup>2</sup>

Lemma 4.2. [7] Let B<sup>1</sup> ∈ H<sup>r</sup>�<sup>s</sup>

i. System (30) is consistent.

ii. R<sup>A</sup>2C<sup>2</sup> <sup>¼</sup> <sup>0</sup>, <sup>C</sup><sup>2</sup> � <sup>A</sup>2C1B†

Since by (10), N†

iii. rank½ �¼ A<sup>2</sup> C<sup>2</sup> rank½ � A<sup>2</sup> , rank

<sup>X</sup> <sup>¼</sup> <sup>C</sup>1B†

In another case,

cdet<sup>i</sup> <sup>H</sup><sup>∗</sup> ð Þ <sup>H</sup> :<sup>i</sup> <sup>b</sup><sup>c</sup>

cdet<sup>i</sup> <sup>H</sup><sup>∗</sup> ð Þ <sup>H</sup> :<sup>i</sup> <sup>v</sup>ð Þ<sup>2</sup>

ð Þ2 <sup>1</sup><sup>j</sup> ;…; v

2 � �

Let in Eq.(1), the matrix A<sup>1</sup> is vanished. Then, we have the system,

<sup>j</sup>: <sup>b</sup><sup>c</sup> ð Þ2 q: � � � � <sup>α</sup>

�

The following lemma is extended to matrices with quaternion entries as well.

C1 B1 � �

<sup>1</sup>B<sup>2</sup> � �N†

<sup>2</sup> <sup>C</sup><sup>2</sup> � <sup>A</sup>2C1B†

where Z<sup>2</sup> and W<sup>2</sup> are the arbitrary matrices over H with appropriate sizes.

<sup>R</sup><sup>B</sup><sup>1</sup> <sup>¼</sup> <sup>N</sup>†

<sup>2</sup>C2N† � <sup>A</sup>2C1B†

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

<sup>2</sup>C2N† � <sup>A</sup>†

The following theorem on determinantal representations of (29) can be proven similar to the

<sup>X</sup><sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> is to be determined. Put <sup>N</sup> <sup>¼</sup> <sup>R</sup><sup>B</sup>1B2. Then, the following statements are equivalent:

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

<sup>1</sup>B<sup>2</sup> � �L<sup>N</sup> <sup>¼</sup> <sup>0</sup>, <sup>C</sup>2L<sup>B</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>.

†

By putting Z2=W2=0, there is the following partial solution of (30),

<sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>C</sup>1B†

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

In this case, the general solution of (30) can be expressed as

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

R<sup>B</sup><sup>1</sup> ¼ ð Þ R<sup>B</sup>1B<sup>2</sup>

<sup>X</sup> <sup>¼</sup> <sup>C</sup>1B†

proof of Theorem 3.1 as well.

:<sup>j</sup> ¼ v

rdet<sup>j</sup> B2B<sup>∗</sup>

Proof. The proof is similar to the proof of Theorem 3.1.

ð Þ2 :t � � � � <sup>β</sup>

:j � � � � <sup>β</sup>

> ð Þ2 nj � �

β , uð Þ<sup>1</sup>

β , gð Þ<sup>2</sup>

α , uð Þ<sup>2</sup>

XB<sup>1</sup> ¼ C1, A2XB<sup>2</sup> ¼ C2:

and uð Þ<sup>2</sup>

iz <sup>≔</sup> <sup>X</sup> β∈ Jr4,nf gi

il <sup>≔</sup> <sup>X</sup> β∈ Jr4,nf gi

:<sup>l</sup> <sup>¼</sup> <sup>u</sup>ð Þ<sup>2</sup>

ql <sup>≔</sup> <sup>X</sup> α∈Ir1,mf gl

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

¼ rank½ � B<sup>1</sup> , rank

<sup>1</sup><sup>l</sup> ;…; <sup>u</sup>ð Þ<sup>2</sup> nl � �

rdet<sup>l</sup> A1A<sup>∗</sup>

C<sup>2</sup> A2C<sup>1</sup> B<sup>2</sup> B<sup>1</sup> � �

, then some simplification of (31) can be derived,

<sup>1</sup>B2N† <sup>þ</sup> <sup>L</sup><sup>A</sup>2W2R<sup>B</sup><sup>1</sup> <sup>þ</sup> <sup>Z</sup>2RNR<sup>B</sup><sup>1</sup> :

1B2N†

2A2C1B†

R<sup>B</sup><sup>1</sup> þ L<sup>A</sup>2W2R<sup>B</sup><sup>1</sup> þ Z2RNR<sup>B</sup><sup>1</sup> , (31)

cdet<sup>i</sup> <sup>H</sup><sup>∗</sup> ð Þ <sup>H</sup> :<sup>i</sup> <sup>b</sup>að Þ<sup>2</sup>

cdet<sup>i</sup> <sup>H</sup><sup>∗</sup> ð Þ <sup>H</sup> :<sup>i</sup> <sup>u</sup>ð Þ<sup>2</sup>

:z � � � � <sup>β</sup>

:l � � � � <sup>β</sup>

<sup>l</sup>: <sup>a</sup>ð Þ<sup>2</sup> q: � � � � <sup>α</sup>

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

¼ rank½ � B<sup>2</sup> B<sup>1</sup> .

: (32)

such that

1 � � β :

β :

α

(30)

$$d\_{\vec{\eta}}^{(2)} \coloneqq \sum\_{\beta \in I\_{\gamma\_2,n}\{i\}} \mathsf{cdet}\_i \Big( \left( \mathbf{A}\_2^\* \mathbf{A}\_2 \right)\_{,i} \Big( \boldsymbol{\uprho}\_{,\dot{\jmath}}^{(2)} \Big) \Big)\_{\boldsymbol{\upbeta}'}^{\beta} \\ \mathop{\mathsf{g}}\_{z\dot{\jmath}}^{(2)} \coloneqq \sum\_{\beta \in I\_{\gamma\_1,n}\{z\}} \mathsf{cdet}\_{z} \Big( \left( \mathbf{B}\_1^\* \mathbf{B}\_1 \right)\_{,z} \Big( \boldsymbol{\uprho}\_{,\dot{\jmath}}^{(2)} \Big) \Big)\_{\boldsymbol{\upbeta}'}^{\beta}$$

and the column-vectors φð Þ<sup>2</sup> :<sup>j</sup> <sup>¼</sup> <sup>φ</sup>ð Þ<sup>2</sup> <sup>1</sup><sup>j</sup> ;…;φð Þ<sup>2</sup> nj � � and <sup>ψ</sup>ð Þ<sup>2</sup> :<sup>j</sup> <sup>¼</sup> <sup>ψ</sup>ð Þ<sup>2</sup> <sup>1</sup><sup>j</sup> ;…; <sup>ψ</sup>ð Þ<sup>2</sup> sj � � such that

$$\boldsymbol{\phi}\_{\boldsymbol{q}\boldsymbol{j}}^{(2)} = \sum\_{\boldsymbol{a}\in I\_{\boldsymbol{\Psi}^{\*}}\{\boldsymbol{j}\}} \mathbf{rdet}\_{\boldsymbol{\up{f}}} \Big( (\mathbf{NN}^{\*})\_{\boldsymbol{\up{f}}} \Big( \mathbf{c}\_{\boldsymbol{q}\cdot}^{(2)} \Big) \Big)\_{\boldsymbol{a}'}^{\boldsymbol{a}} \,\boldsymbol{\up{y}}\_{\boldsymbol{u}\boldsymbol{j}}^{(2)} \coloneqq \sum\_{\boldsymbol{a}\in I\_{\boldsymbol{\Psi}^{\*}}\{\boldsymbol{j}\}} \mathbf{rdet}\_{\boldsymbol{\up{f}}} \Big( (\mathbf{NN}^{\*})\_{\boldsymbol{\up{f}}} \Big( \boldsymbol{b}\_{\boldsymbol{u}\cdot}^{(2)} \Big) \Big)\_{\boldsymbol{a}}^{\boldsymbol{a}} \,\boldsymbol{\up{y}}\_{\boldsymbol{a}\boldsymbol{b}}^{(2)}$$

Now, suppose that the both equations of (1) are one-sided. Let in Eq.(1), the matrices B<sup>1</sup> and A<sup>2</sup> are vanished. Then, we have the system

$$\begin{cases} \mathbf{A}\_1 \mathbf{X} = \mathbf{C}\_1, \\ \mathbf{X} \mathbf{B}\_2 = \mathbf{C}\_2. \end{cases} \tag{33}$$

The following lemma is extended to matrices with quaternion entries.

Lemma 4.3. [31] Let A<sup>1</sup> ∈ H<sup>m</sup>�<sup>n</sup>, B<sup>2</sup> ∈ H<sup>r</sup>�<sup>p</sup>, C<sup>1</sup> ∈ H<sup>m</sup>�<sup>r</sup> , and C<sup>2</sup> ∈ H<sup>n</sup>�<sup>p</sup> be given and X∈ H<sup>n</sup>�<sup>r</sup> is to be determined. Then, the system (33) is consistent if and only if R<sup>A</sup>1C<sup>1</sup> ¼ 0, C2L<sup>B</sup><sup>2</sup> ¼ 0, and A1C2=C1B2. Under these conditions, the general solution to (33) can be established as

$$\mathbf{X} = \mathbf{A}\_1^\dagger \mathbf{C}\_1 + \mathbf{L}\_{A\_1} \mathbf{C}\_2 \mathbf{B}\_2^\dagger + \mathbf{L}\_{A\_1} \mathbf{U} \mathbf{R}\_{B\_2},\tag{34}$$

Let in Eq.(1), the matrices B1 and B2 are vanished. Then, we have the system

Lemma 4.4. [7] Suppose that A<sup>1</sup> ∈ H<sup>m</sup>�<sup>n</sup>, C<sup>1</sup> ∈ H<sup>m</sup>�<sup>r</sup>

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

<sup>i</sup> <sup>C</sup><sup>i</sup> <sup>¼</sup> <sup>C</sup>i, for all i <sup>¼</sup> <sup>1</sup>, <sup>2</sup> and T A†

to (38) can be established as

A2A†

Theorem 4.4. Let A<sup>1</sup> ¼ a

, and there exist A†

ij � � <sup>∈</sup> <sup>H</sup><sup>n</sup>�n. Then, <sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup><sup>0</sup>

P

P

<sup>L</sup><sup>A</sup><sup>1</sup> g ¼ <sup>r</sup>3. Denote <sup>A</sup><sup>∗</sup>

x0 ij ¼

� Xn l¼1

ð Þ1 :<sup>j</sup> , <sup>b</sup><sup>c</sup> ð Þ2 :<sup>j</sup> , and a

AiA†

H†

H<sup>k</sup>�<sup>r</sup>

ba ð Þ2

where <sup>b</sup><sup>c</sup>

<sup>C</sup><sup>2</sup> � <sup>H</sup>†

considered

�

<sup>2</sup>C<sup>2</sup> � <sup>A</sup>†

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

<sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>A</sup>†

ð Þ1

<sup>1</sup>C1≕Cb<sup>1</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup>

<sup>β</sup>∈Jr1,nf g<sup>i</sup> cdet<sup>i</sup> <sup>A</sup><sup>∗</sup>

P

P

Proof. The proof is similar to the proof of Theorem 3.1.

bð Þ2

ð Þ1 , †

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

<sup>β</sup><sup>∈</sup> Jr3,nf g<sup>i</sup> cdet<sup>i</sup> <sup>H</sup><sup>∗</sup> ð Þ <sup>H</sup> :<sup>i</sup> <sup>b</sup>að Þ<sup>2</sup>

<sup>β</sup>∈Jr3,n <sup>H</sup><sup>∗</sup> j j <sup>H</sup> <sup>β</sup>

<sup>1</sup> ¼ a

In the following theorem, we give the determinantal representations of (40).

ij � �<sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>n</sup> <sup>r</sup><sup>1</sup> , <sup>A</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup>

ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>m</sup>, <sup>H</sup>†

ð Þ1 ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup>

<sup>1</sup>A<sup>1</sup> � �

<sup>1</sup>A<sup>1</sup> � � � � β β

β

:<sup>i</sup> <sup>b</sup><sup>c</sup> ð Þ1 :j � � � � <sup>β</sup>

:l � � � � <sup>β</sup>

where Y is an arbitrary matrix over H with an appropriate size.

A1X ¼ C1, A2X ¼ C2:

<sup>X</sup><sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> is unknown, <sup>H</sup> <sup>¼</sup> <sup>A</sup>2L<sup>A</sup><sup>1</sup> , <sup>T</sup> <sup>¼</sup> <sup>R</sup>HA2. Then, the system (38) is consistent if and only if

<sup>1</sup>C<sup>1</sup>

A<sup>2</sup> A†

Using (10) and the consistency conditions, we simplify (39) accordingly, <sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>A</sup>†

<sup>1</sup>C<sup>1</sup> <sup>þ</sup> <sup>H</sup>†

<sup>2</sup>C<sup>2</sup> � <sup>A</sup>†

<sup>C</sup><sup>2</sup> � <sup>H</sup>†

ð Þ2

<sup>2</sup> <sup>¼</sup> <sup>h</sup>†

β

β

� P

þ

P

<sup>1</sup>C<sup>1</sup>

<sup>1</sup>C<sup>1</sup> þ L<sup>A</sup>1LHY: Consequently, the following partial solution of (39) will be

A2A†

ij � �<sup>∈</sup> <sup>H</sup><sup>k</sup>�<sup>n</sup> <sup>r</sup><sup>2</sup> , <sup>C</sup><sup>1</sup> <sup>¼</sup> <sup>c</sup>

ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>s</sup>

, <sup>H</sup><sup>∗</sup>C2≕Cb<sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup>

ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> possesses the following determinantal representation,

(38)

17

<sup>1</sup>C1þ

, A<sup>2</sup> ∈ H<sup>k</sup>�n, and C<sup>2</sup> ∈ H<sup>k</sup>�<sup>r</sup> are known and

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

<sup>1</sup>C1: (40)

. Let rankH ¼ min rank f A2;rank

, and H<sup>∗</sup>

ð Þ2 :j � � � � <sup>β</sup>

β

:<sup>l</sup> <sup>b</sup><sup>c</sup> ð Þ1 :j � � � � <sup>β</sup>

β

β

,

, C<sup>2</sup> ¼ c

ð Þ2 ij � � <sup>∈</sup>

A2≕Ab <sup>2</sup> ¼

(41)

� � <sup>þ</sup> <sup>L</sup><sup>A</sup>1LHY, (39)

ð Þ1 ij � � <sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>r</sup>

ð Þ2 ij � � <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup>

<sup>β</sup>∈Jr3,nf g<sup>i</sup> cdet<sup>i</sup> <sup>H</sup><sup>∗</sup> ð Þ <sup>H</sup> :<sup>i</sup> <sup>b</sup><sup>c</sup>

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

<sup>β</sup>∈Jr3,n <sup>H</sup><sup>∗</sup> j j <sup>H</sup> <sup>β</sup>

<sup>1</sup>A<sup>1</sup> � �

<sup>1</sup>A<sup>1</sup> � � � � β β

P

<sup>β</sup>∈Jr1,nf g<sup>l</sup> cdet<sup>l</sup> <sup>A</sup><sup>∗</sup>

P

:<sup>j</sup> are the jth columns of the matrices Cb1, Cb 2, and Ab 2, respectively.

� � <sup>¼</sup> <sup>0</sup>. Under these conditions, the general solution

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

where U is a free matrix over H with a suitable shape.

Due to the consistence conditions, Eq. (34) can be expressed as follows:

$$\begin{split} \mathbf{X} &= \mathbf{C}\_{2}\mathbf{B}\_{2}^{\dagger} + \mathbf{A}\_{1}^{\dagger} \left( \mathbf{C}\_{1} - \mathbf{A}\_{1}\mathbf{C}\_{2}\mathbf{B}\_{2}^{\dagger} \right) + \mathbf{L}\_{A\_{1}}\mathbf{U}\mathbf{R}\_{\mathcal{B}\_{2}} \\ &= \mathbf{C}\_{2}\mathbf{B}\_{2}^{\dagger} + \mathbf{A}\_{1}^{\dagger} \left( \mathbf{C}\_{1} - \mathbf{C}\_{1}\mathbf{B}\_{2}\mathbf{B}\_{2}^{\dagger} \right) + \mathbf{L}\_{A\_{1}}\mathbf{U}\mathbf{R}\_{\mathcal{B}\_{2}} = \mathbf{C}\_{2}\mathbf{B}\_{2}^{\dagger} + \mathbf{A}\_{1}^{\dagger}\mathbf{C}\_{1}\mathbf{R}\_{\mathcal{B}\_{2}} + \mathbf{L}\_{A\_{1}}\mathbf{U}\mathbf{R}\_{\mathcal{B}\_{2}}. \end{split}$$

Consequently, the partial solution X<sup>0</sup> to (33) is given by

$$\mathbf{X}^{0} = \mathbf{A}\_{1}^{\dagger}\mathbf{C}\_{1} + \mathbf{L}\_{A\_{1}}\mathbf{C}\_{2}\mathbf{B}\_{2}^{\dagger} \tag{35}$$

or

$$\mathbf{X}^{0} = \mathbf{C}\_{2}\mathbf{B}\_{2}^{\dagger} + \mathbf{A}\_{1}^{\dagger}\mathbf{C}\_{1}\mathbf{R}\_{\partial\_{2}}.\tag{36}$$

Due to the expression (35), the following theorem can be proven similar to the proof of Theorem 3.1.

Theorem 4.3. Let A<sup>1</sup> ¼ a ð Þ1 ij � �<sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>n</sup> <sup>r</sup><sup>1</sup> , <sup>B</sup><sup>2</sup> <sup>¼</sup> <sup>b</sup> ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>r</sup>�<sup>p</sup> <sup>r</sup><sup>2</sup> , <sup>C</sup><sup>1</sup> <sup>¼</sup> <sup>c</sup> ð Þ1 ij � �<sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>r</sup> , and C<sup>2</sup> ¼ c ð Þ2 ij � � ∈ H<sup>n</sup>�<sup>r</sup> , and there exist A† <sup>1</sup> ¼ a ð Þ1 ,† ij � � <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>m</sup>, <sup>B</sup>† <sup>2</sup> ¼ b ð Þ2 , † ij � �<sup>∈</sup> <sup>H</sup><sup>p</sup>�<sup>r</sup> , and <sup>L</sup><sup>A</sup><sup>1</sup> <sup>¼</sup> <sup>I</sup> � <sup>A</sup>† <sup>1</sup>A1<sup>≕</sup> lij � � ∈ H<sup>n</sup>�n. Denote A<sup>∗</sup> <sup>1</sup>C1≕Cb<sup>1</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup> ð Þ1 ij � � <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> and <sup>L</sup><sup>A</sup>1C2B<sup>∗</sup> <sup>2</sup>≕Cb<sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup> ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> . Then, the partial solution (35), <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>s</sup> , possesses the following determinantal representation,

$$\mathbf{x}\_{\boldsymbol{\hat{\eta}}}^{0} = \frac{\sum\_{\boldsymbol{\beta} \in I\_{\boldsymbol{\gamma}\_{1}, \boldsymbol{\epsilon}} \{\boldsymbol{\beta}\}} \mathbf{c} \mathbf{det}\_{i} \Big( \left( \mathbf{A}\_{1}^{\ast} \mathbf{A}\_{1} \right)\_{\boldsymbol{\beta}} \Big( \hat{\mathbf{c}}\_{\boldsymbol{\cdot}, \boldsymbol{\epsilon}}^{(1)} \Big) \Big)\_{\boldsymbol{\beta}}^{\beta}}{\sum\_{\boldsymbol{\beta} \in I\_{\boldsymbol{\gamma}\_{1}, \boldsymbol{\epsilon}}} \left| \mathbf{A}\_{1}^{\ast} \mathbf{A}\_{1} \right|\_{\boldsymbol{\beta}}^{\beta}} + \frac{\sum\_{\boldsymbol{\alpha} \in I\_{\boldsymbol{\alpha}\_{2}, \boldsymbol{\epsilon}} \{\boldsymbol{\beta}\}} \mathbf{r} \mathbf{det}\_{i} \Big( \left( \mathbf{B}\_{2} \mathbf{B}\_{2}^{\ast} \right)\_{\boldsymbol{\beta}} \Big( \hat{\mathbf{c}}\_{i}^{(2)} \Big) \Big)\_{\boldsymbol{\alpha}}^{\alpha}}{\sum\_{\boldsymbol{\alpha} \in I\_{\boldsymbol{\alpha}\_{2}, \boldsymbol{\epsilon}}} \left| \mathbf{B}\_{2} \mathbf{B}\_{2}^{\ast} \right|\_{\boldsymbol{\alpha}}^{\alpha}} \right) \tag{37}$$

where <sup>b</sup><sup>c</sup> ð Þ1 :<sup>j</sup> is the <sup>j</sup>th column of <sup>C</sup>b<sup>1</sup> and <sup>b</sup><sup>c</sup> ð Þ2 <sup>i</sup>: is the ith row of Cb 2.

Remark 4.1. In accordance to the expression (36), we obtain the same representations, but with the denotations, C2B<sup>∗</sup> <sup>2</sup>≕Cb<sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup> ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> and <sup>A</sup><sup>∗</sup> <sup>1</sup>C1R<sup>B</sup>2≕Cb<sup>1</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup> ð Þ2 ij � � <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> .

Let in Eq.(1), the matrices B1 and B2 are vanished. Then, we have the system

Lemma 4.3. [31] Let A<sup>1</sup> ∈ H<sup>m</sup>�<sup>n</sup>, B<sup>2</sup> ∈ H<sup>r</sup>�<sup>p</sup>, C<sup>1</sup> ∈ H<sup>m</sup>�<sup>r</sup>

where U is a free matrix over H with a suitable shape.

<sup>X</sup> <sup>¼</sup> <sup>C</sup>2B†

16 Matrix Theory-Applications and Theorems

or

Theorem 3.1.

∈ H<sup>n</sup>�<sup>r</sup>

where <sup>b</sup><sup>c</sup>

Theorem 4.3. Let A<sup>1</sup> ¼ a

∈ H<sup>n</sup>�n. Denote A<sup>∗</sup>

solution (35), <sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup><sup>0</sup>

x0 ij ¼

ð Þ1

denotations, C2B<sup>∗</sup>

, and there exist A†

P

<sup>¼</sup> <sup>C</sup>2B†

<sup>2</sup> <sup>þ</sup> <sup>A</sup>†

<sup>2</sup> <sup>þ</sup> <sup>A</sup>†

be determined. Then, the system (33) is consistent if and only if R<sup>A</sup>1C<sup>1</sup> ¼ 0, C2L<sup>B</sup><sup>2</sup> ¼ 0, and

<sup>1</sup>C<sup>1</sup> <sup>þ</sup> <sup>L</sup><sup>A</sup>1C2B†

A1C2=C1B2. Under these conditions, the general solution to (33) can be established as

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

Due to the consistence conditions, Eq. (34) can be expressed as follows:

2 � � <sup>þ</sup> <sup>L</sup><sup>A</sup>1UR<sup>B</sup><sup>2</sup>

2 � � <sup>þ</sup> <sup>L</sup><sup>A</sup>1UR<sup>B</sup><sup>2</sup> <sup>¼</sup> <sup>C</sup>2B†

<sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>A</sup>†

<sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>C</sup>2B†

<sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>n</sup> <sup>r</sup><sup>1</sup> , <sup>B</sup><sup>2</sup> <sup>¼</sup> <sup>b</sup>

ð Þ1 ,† ij � �

> <sup>1</sup>A<sup>1</sup> � �

<sup>1</sup>A<sup>1</sup> � � � � β β

:<sup>i</sup> <sup>b</sup><sup>c</sup> ð Þ1 :j � � � � <sup>β</sup>

ð Þ2

∈ H<sup>n</sup>�<sup>r</sup> and A<sup>∗</sup>

ð Þ1 ij � � <sup>1</sup>C<sup>1</sup> <sup>þ</sup> <sup>L</sup><sup>A</sup>1C2B†

<sup>2</sup> <sup>þ</sup> <sup>A</sup>†

Due to the expression (35), the following theorem can be proven similar to the proof of

ð Þ2 ij � �

<sup>2</sup> ¼ b

∈ H<sup>n</sup>�<sup>m</sup>, B†

∈ H<sup>n</sup>�<sup>r</sup> and L<sup>A</sup>1C2B<sup>∗</sup>

β

Remark 4.1. In accordance to the expression (36), we obtain the same representations, but with the

þ P

<sup>i</sup>: is the ith row of Cb 2.

<sup>1</sup>C1R<sup>B</sup>2≕Cb<sup>1</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup>

<sup>1</sup> <sup>C</sup><sup>1</sup> � <sup>A</sup>1C2B†

<sup>1</sup> <sup>C</sup><sup>1</sup> � <sup>C</sup>1B2B†

Consequently, the partial solution X<sup>0</sup> to (33) is given by

ð Þ1 ij � �

<sup>1</sup>C1≕Cb<sup>1</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup>

∈ H<sup>n</sup>�<sup>s</sup>

<sup>β</sup>∈Jr1,nf g<sup>i</sup> cdet<sup>i</sup> <sup>A</sup><sup>∗</sup>

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

ð Þ2 ij � �

P

:<sup>j</sup> is the <sup>j</sup>th column of <sup>C</sup>b<sup>1</sup> and <sup>b</sup><sup>c</sup>

<sup>2</sup>≕Cb<sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup>

ij � � <sup>1</sup> ¼ a

, and C<sup>2</sup> ∈ H<sup>n</sup>�<sup>p</sup> be given and X∈ H<sup>n</sup>�<sup>r</sup> is to

<sup>2</sup> þ L<sup>A</sup>1UR<sup>B</sup><sup>2</sup> , (34)

<sup>1</sup>C1R<sup>B</sup><sup>2</sup> þ L<sup>A</sup>1UR<sup>B</sup><sup>2</sup> ,

<sup>1</sup>C1R<sup>B</sup><sup>2</sup> : (36)

∈ H<sup>m</sup>�<sup>r</sup>

∈ H<sup>n</sup>�<sup>r</sup>

2 � �

2

<sup>P</sup> <sup>α</sup> <sup>α</sup> <sup>∈</sup>Ir2,r <sup>B</sup>2B<sup>∗</sup>

� � � � α α

∈ H<sup>n</sup>�<sup>r</sup> .

<sup>j</sup>: <sup>b</sup><sup>c</sup> ð Þ2 i: � � � � <sup>α</sup>

, and <sup>L</sup><sup>A</sup><sup>1</sup> <sup>¼</sup> <sup>I</sup> � <sup>A</sup>†

, and C<sup>2</sup> ¼ c

. Then, the partial

ð Þ2 ij � �

<sup>1</sup>A1≕ lij � �

, (37)

ð Þ1 ij � �

ð Þ2 ij � �

∈ H<sup>p</sup>�<sup>r</sup>

<sup>α</sup><sup>∈</sup> Ir2,rf g<sup>j</sup> rdet<sup>j</sup> <sup>B</sup>2B<sup>∗</sup>

ð Þ2 ij � �

<sup>2</sup>, (35)

<sup>2</sup> <sup>þ</sup> <sup>A</sup>†

<sup>∈</sup> <sup>H</sup><sup>r</sup>�<sup>p</sup> <sup>r</sup><sup>2</sup> , <sup>C</sup><sup>1</sup> <sup>¼</sup> <sup>c</sup>

<sup>2</sup>≕Cb<sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup>

ð Þ2 , † ij � �

, possesses the following determinantal representation,

$$\begin{cases} \mathbf{A}\_1 \mathbf{X} = \mathbf{C}\_1, \\ \mathbf{A}\_2 \mathbf{X} = \mathbf{C}\_2. \end{cases} \tag{38}$$

Lemma 4.4. [7] Suppose that A<sup>1</sup> ∈ H<sup>m</sup>�<sup>n</sup>, C<sup>1</sup> ∈ H<sup>m</sup>�<sup>r</sup> , A<sup>2</sup> ∈ H<sup>k</sup>�n, and C<sup>2</sup> ∈ H<sup>k</sup>�<sup>r</sup> are known and <sup>X</sup><sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> is unknown, <sup>H</sup> <sup>¼</sup> <sup>A</sup>2L<sup>A</sup><sup>1</sup> , <sup>T</sup> <sup>¼</sup> <sup>R</sup>HA2. Then, the system (38) is consistent if and only if AiA† <sup>i</sup> <sup>C</sup><sup>i</sup> <sup>¼</sup> <sup>C</sup>i, for all i <sup>¼</sup> <sup>1</sup>, <sup>2</sup> and T A† <sup>2</sup>C<sup>2</sup> � <sup>A</sup>† <sup>1</sup>C<sup>1</sup> � � <sup>¼</sup> <sup>0</sup>. Under these conditions, the general solution to (38) can be established as

$$\mathbf{X} = \mathbf{A}\_1^\dagger \mathbf{C}\_1 + \mathbf{L}\_{A\_1} \mathbf{H}^\dagger \mathbf{A}\_2 \left(\mathbf{A}\_2^\dagger \mathbf{C}\_2 - \mathbf{A}\_1^\dagger \mathbf{C}\_1\right) + \mathbf{L}\_{A\_1} \mathbf{L}\_H \mathbf{Y},\tag{39}$$

where Y is an arbitrary matrix over H with an appropriate size.

Using (10) and the consistency conditions, we simplify (39) accordingly, <sup>X</sup><sup>0</sup> <sup>¼</sup> <sup>A</sup>† <sup>1</sup>C1þ H† <sup>C</sup><sup>2</sup> � <sup>H</sup>† A2A† <sup>1</sup>C<sup>1</sup> þ L<sup>A</sup>1LHY: Consequently, the following partial solution of (39) will be considered

$$\mathbf{X}^{0} = \mathbf{A}\_{1}^{\dagger}\mathbf{C}\_{1} + \mathbf{H}^{\dagger}\mathbf{C}\_{2} - \mathbf{H}^{\dagger}\mathbf{A}\_{2}\mathbf{A}\_{1}^{\dagger}\mathbf{C}\_{1}.\tag{40}$$

In the following theorem, we give the determinantal representations of (40).

Theorem 4.4. Let A<sup>1</sup> ¼ a ð Þ1 ij � �<sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>n</sup> <sup>r</sup><sup>1</sup> , <sup>A</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup> ð Þ2 ij � �<sup>∈</sup> <sup>H</sup><sup>k</sup>�<sup>n</sup> <sup>r</sup><sup>2</sup> , <sup>C</sup><sup>1</sup> <sup>¼</sup> <sup>c</sup> ð Þ1 ij � � <sup>∈</sup> <sup>H</sup><sup>m</sup>�<sup>r</sup> , C<sup>2</sup> ¼ c ð Þ2 ij � � <sup>∈</sup> H<sup>k</sup>�<sup>r</sup> , and there exist A† <sup>1</sup> ¼ a ð Þ1 , † ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>m</sup>, <sup>H</sup>† <sup>2</sup> <sup>¼</sup> <sup>h</sup>† ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>s</sup> . Let rankH ¼ min rank f A2;rank <sup>L</sup><sup>A</sup><sup>1</sup> g ¼ <sup>r</sup>3. Denote <sup>A</sup><sup>∗</sup> <sup>1</sup>C1≕Cb<sup>1</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup> ð Þ1 ij � �<sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> , <sup>H</sup><sup>∗</sup>C2≕Cb<sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>c</sup> ð Þ2 ij � � <sup>∈</sup> <sup>H</sup><sup>n</sup>�<sup>r</sup> , and <sup>H</sup><sup>∗</sup>A2≕A<sup>b</sup> <sup>2</sup> <sup>¼</sup> ba ð Þ2 ij � � <sup>∈</sup> <sup>H</sup><sup>n</sup>�n. 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> possesses the following determinantal representation,

x0 ij ¼ P <sup>β</sup>∈Jr1,nf g<sup>i</sup> cdet<sup>i</sup> <sup>A</sup><sup>∗</sup> <sup>1</sup>A<sup>1</sup> � � :<sup>i</sup> <sup>b</sup><sup>c</sup> ð Þ1 :j � � � � <sup>β</sup> β P <sup>β</sup>∈Jr1,n <sup>A</sup><sup>∗</sup> <sup>1</sup>A<sup>1</sup> � � � � β β þ P <sup>β</sup>∈Jr3,nf g<sup>i</sup> cdet<sup>i</sup> <sup>H</sup><sup>∗</sup> ð Þ <sup>H</sup> :<sup>i</sup> <sup>b</sup><sup>c</sup> ð Þ2 :j � � � � <sup>β</sup> β P <sup>β</sup>∈Jr3,n <sup>H</sup><sup>∗</sup> j j <sup>H</sup> <sup>β</sup> β � Xn l¼1 P <sup>β</sup><sup>∈</sup> Jr3,nf g<sup>i</sup> cdet<sup>i</sup> <sup>H</sup><sup>∗</sup> ð Þ <sup>H</sup> :<sup>i</sup> <sup>b</sup>að Þ<sup>2</sup> :l � � � � <sup>β</sup> β P <sup>β</sup>∈Jr3,n <sup>H</sup><sup>∗</sup> j j <sup>H</sup> <sup>β</sup> β � P <sup>β</sup>∈Jr1,nf g<sup>l</sup> cdet<sup>l</sup> <sup>A</sup><sup>∗</sup> <sup>1</sup>A<sup>1</sup> � � :<sup>l</sup> <sup>b</sup><sup>c</sup> ð Þ1 :j � � � � <sup>β</sup> β P <sup>β</sup><sup>∈</sup> Jr1,n <sup>A</sup><sup>∗</sup> <sup>1</sup>A<sup>1</sup> � � � � β β , (41)

where <sup>b</sup><sup>c</sup> ð Þ1 :<sup>j</sup> , <sup>b</sup><sup>c</sup> ð Þ2 :<sup>j</sup> , and a bð Þ2 :<sup>j</sup> are the jth columns of the matrices Cb1, Cb 2, and Ab 2, respectively. Proof. The proof is similar to the proof of Theorem 3.1.
