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<title>New sufficient conditions for the solvability <br />of a new class of Sylvester-like <br />absolute value matrix equations: New sufficient conditions for the solvability <br />of a new class of Sylvester-like <br />absolute value matrix equations</title>
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<h1>New sufficient conditions for the solvability <br />of a new class of Sylvester-like <br />absolute value matrix equations</h1>
<p class="authors">
<span class="author">Shubham Kumar\(^\ast \), Deepmala\(^\dag \) Roshan Lal Keshtwal\(^\ddag \)</span>
</p>
<p class="date">April 10, 2023; accepted: September 28, 2023; published online: December 22, 2023.</p>
<p class="thanks">\(^\ast \)The research work of Shubham Kumar was supported by the Ministry of Education, Government of India, through Graduate Aptitude Test in Engineering (GATE) fellowship registration No. MA19S43033021. <br />\(^\ast \)\(^\dag \) Mathematics Discipline, PDPM-Indian Institute of Information Technology, Design and Manufacturing, Jabalpur, Madhya Pradesh, India, e-mails: <span class="tt">\(^\ast \)shub.srma@gmail.com</span>, <span class="tt">\(^\dag \)dmrai23@gmail.com</span> <br />\(^\ddag \)VSKC Government Postgraduate College Dakpathar, Dehradun, Uttarakhand, India, e-mail: <span class="tt">rlkeshtwal@gmail.com</span></p>
</div>



<div class="abstract"> In this article, some new sufficient conditions for the unique solvability of a new class of Sylvester-like absolute value matrix equations \(AXB - \vert CXD \vert =F\) are obtained. This work is distinct from the published work by Li [Journal of Optimization Theory and Application, 195(2), 2022]. Some new conditions were also obtained, which were not covered by Li. We also give an example in support of our result. </div>
<p><b class="bfseries">MSC. </b>15A06, 90C05, 90C30. </p>
<p><b class="bfseries">Keywords. </b>New Sylvester-like absolute value matrix equation, sufficient condition, unique solution. </p>
<h1 id="a0000000002">1 Introduction</h1>
<p>Recently, Li <span class="cite">
	[
	<a href="index.html#Li 2022" >8</a>
	]
</span> introduced the following new class of Sylvester-like absolute value matrix equations (AVME) </p>
<div class="equation" id="equ1">
<p>
  <div class="equation_content">
    \begin{equation}  \label{equ1} AXB-|CXD|=F, \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> where \(A,B,C,D,F\) \(\in \) \(\mathbb {R}^{n \times n}\) are given and \(X\) \(\in \) \(\mathbb {R}^{n \times n}\) to be determined. Eq. (<a href="index.html#equ1">1</a>) is a special case of the following new generalized absolute value equations (NGAVE) </p>
<div class="equation" id="equ2">
<p>
  <div class="equation_content">
    \begin{equation}  \label{equ2} Ax-|Cx|=f, \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> with \(A,C\) \(\in \) \(\mathbb {R}^{n \times n},\) \(f\) \(\in \) \(\mathbb {R}^{n}\) are known and \(x\) \(\in \) \(\mathbb {R}^{n}\) is unknown. </p>
<p>The generalized absolute value equations (GAVE) is defined as </p>
<div class="equation" id="equ3">
<p>
  <div class="equation_content">
    \begin{equation}  \label{equ3} Ax-B|x|=f. \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p>The generalized absolute value matrix equations (GAVME) is a generalization version of the GAVE (<a href="index.html#equ3">3</a>) and is defined as </p>
<div class="equation" id="equ4">
<p>
  <div class="equation_content">
    \begin{equation}  \label{equ4} AX+B|X|=F. \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p>The Sylvester-like AVME is defined as </p>
<div class="equation" id="equ5">
<p>
  <div class="equation_content">
    \begin{equation}  \label{equ5} AXB+C|X|D=F. \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p> The new class of Sylvester-like AVME (<a href="index.html#equ1">1</a>) is quite different from the Sylvester-like AVME (<a href="index.html#equ5">5</a>). </p>
<p>The absolute value equations is powerful tools in the field of optimization, complementarity problems, convex quadratic programming and linear programming. For more about the absolute value equations, one may refer to (<p><span class="cite">
	[
	<a href="index.html#cottle 1992" >1</a>
	, 
	<a href="index.html#Mangasarian 2006" >9</a>
	, 
	<a href="index.html#KG Murty 1992" >10</a>
	, 
	<a href="index.html#Jiri Rohn 2004" >11</a>
	, 
	<a href="index.html#Shi Liang Wu 2021" >19</a>
	]
</span> and references therein). </p>
<p>In 2021, the NGAVE (<a href="index.html#equ2">2</a>) was first considered by Wu <span class="cite">
	[
	<a href="index.html#Shi Liang Wu 2021" >19</a>
	]
</span> and discussed its different conditions for a unique solution. In 2020, Dehghan <i class="it">et al.</i> <span class="cite">
	[
	<a href="index.html#Dehghan 2020" >2</a>
	]
</span> first considered the generalized absolute value matrix equations (<a href="index.html#equ4">4</a>) and provided a matrix multi-splitting Picard-iterative method for solving the GAVME. In 2022, Kumar <i class="it">et al.</i> <span class="cite">
	[
	<a href="index.html#Sharma 2022i" >5</a>
	, 
	<a href="index.html#Sharma 2022iii" >6</a>
	]
</span> provided two new conditions to ensure the unique solvability of the GAVME, the condition of Kumar <i class="it">et al.</i> <span class="cite">
	[
	<a href="index.html#Sharma 2022iii" >6</a>
	]
</span> is superior to the conditions of Xie <span class="cite">
	[
	<a href="index.html#Xie 2021" >20</a>
	]
</span> and Dehghan <i class="it">et al.</i> <span class="cite">
	[
	<a href="index.html#Dehghan 2020" >2</a>
	]
</span>. In 2022, Tang <i class="it">et al.</i> <span class="cite">
	[
	<a href="index.html#Tang 2022" >17</a>
	]
</span> further discussed the unique solvability of the GAVME and provided a Picard-type method for the solution of the GAVME. In 2021, Hashemi <span class="cite">
	[
	<a href="index.html#Hasemi 2021" >3</a>
	]
</span> first considered the Sylvester-like absolute value matrix equations (<a href="index.html#equ5">5</a>) and discussed its unique solvability conditions. Wang <i class="it">et al.</i> <span class="cite">
	[
	<a href="index.html#Wang 2021" >18</a>
	]
</span> provided new unique solvability conditions for the Sylvester-like AVME (<a href="index.html#equ5">5</a>), which are different work from the Hashemi <span class="cite">
	[
	<a href="index.html#Hasemi 2021" >3</a>
	]
</span>. Inspired by the above works on different types of matrix equations, Li <span class="cite">
	[
	<a href="index.html#Li 2022" >8</a>
	]
</span> first considered the new class of Sylvester-like AVME (<a href="index.html#equ1">1</a>) and provided unique solvability conditions for (<a href="index.html#equ1">1</a>). </p>
<p>In this article, we further discussed the unique solvability of the new class of Sylvester-like AVME (<a href="index.html#equ1">1</a>). As it has non-differentiable and non-linear terms, studying the new class of Sylvester-like AVME is exciting and challenging. The Sylvester-like absolute value matrix equations have many uses in the field of interval matrix equations <span class="cite">
	[
	<a href="index.html#Neumaier 1990" >13</a>
	, 
	<a href="index.html#Seif 1994" >14</a>
	]
</span> and robust control <span class="cite">
	[
	<a href="index.html#Shashikhin 2002" >15</a>
	]
</span> and so on. </p>
</p>
<h4 id="a0000000003">Notation.</h4>
<p> We will denote \(\hat{D}\) = \(\operatorname {diag}(\hat{d_i})\) with 0 \(\le \) \(\hat{d_i}\) \(\le 1\) is a diagonal matrix. \(\sigma (.)\), \(\sigma _{\max }(.)\) and \(\sigma _{\min }(.)\) denote singular value, maximum singular value and minimum singular value, respectively. For the determinant of a matrix, we will use det(.), and \(\rho (.)\) is used for the spectral radius of a matrix. </p>
<p>The remainder of this paper is structured in the following manner: <a href="index.html#section 2">section 2</a> contains some useful results. In <a href="index.html#section 3">section 3</a>, we obtain the unique solution condition for the new class of Sylvester-like AVME (<a href="index.html#equ1">1</a>). A numerical example in support of our results is provided in <a href="index.html#section 4">section 4</a>, and we conclude our discussion in <a href="index.html#section 5">section 5</a>. </p>
<h1 id="section 2">2 Preliminaries</h1>
<p>  In this section, we recall some definitions, lemmas and theorems for further use. </p>
<p><div class="definition_thmwrapper " id="a0000000004">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">1</span>
    <span class="definition_thmtitle"><span class="cite">
	[
	<a href="index.html#Gowda and Sznajder 1995" >16</a>
	]
</span></span>
  </div>
  <div class="definition_thmcontent">
  <p> Let \(\mathcal{M} =\{ M_1,M_2\} \) denote the set of matrices with \(M_1,M_2\) \(\in \) \(\mathbb {R}^{n \times n}\). A matrix R \(\in \) \(\mathbb {R}^{n \times n}\) is called a row (or column) representative of \(\mathcal{M}\), if \(R_{j.}\) \(\in \) \(\{ (M_{1})_{j.},(M_{2})_{j.}\} \) (or \(R_{.j}\) \(\in \) \(\{ (M_{1})_{.j},(M_{2})_{.j}\} \)) j=1,2,…,n, where \(R_{j.}\), \((M_{1})_{j.}\), and \((M_{2})_{j.}\) (or \(R_{.j}\), \((M_{1})_{.j}\), and \((M_{2})_{.j}\)) denote the \(j^{th}\) row (or column) of R, \(M_{1}\) and \(M_{2}\), respectively. </p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="a0000000005">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2</span>
    <span class="definition_thmtitle"><span class="cite">
	[
	<a href="index.html#Gowda and Sznajder 1995" >16</a>
	]
</span></span>
  </div>
  <div class="definition_thmcontent">
  <p> The set \(\mathcal{M}\) holds the row (or column) \(\mathcal{W}\)-property if the determinants of all row (or column) representative matrices of \(\mathcal{M}\) are positive. </p>

  </div>
</div> <div class="definition_thmwrapper " id="a0000000006">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">3</span>
    <span class="definition_thmtitle"><span class="cite">
	[
	<a href="index.html#Jiri Rohn 2009" >12</a>
	]
</span></span>
  </div>
  <div class="definition_thmcontent">
  <p> For given matrices \(A_{C}, V \in \mathbb {R}^{n \times n}\), \(V \geq 0,\) the set of matrices \(\mathbb {A} =\{  A : \vert A -A_{C} \vert \leq V \} ,\) is known as interval matrix. An interval matrix \(\mathbb {A}\) is regular if each \(A \in \mathbb {A}\) is invertible. </p>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="a0000000007">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">4</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="index.html#Horn 1994" >4</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p> The following results are hold for the square matrices \(A, B, C,\) \( D \in \mathbb {R}^{n \times n}:\) </p>
<p><span class="rm">(i)</span> \(\sigma (A \otimes B) = \sigma (A)\sigma (B).\) </p>
<p><span class="rm">(ii)</span> \(\rho (A \otimes B) = \rho (A)\rho (B).\) </p>
<p><span class="rm">(iii)</span> \((\vert A \otimes B \vert ) = \vert A \vert \otimes \vert B \vert .\) </p>
<p><span class="rm">(iv)</span> \(( A \otimes B ) ( C \otimes D ) = ( AC \otimes BD ).\) </p>
<p><span class="rm">(v)</span> \( vec( ABC ) = ( C^{T} \otimes A )vec(B).\) </p>
<p><span class="rm">(vi)</span> For non-singular matrices A and B, \(( A \otimes B )^{-1} = A^{-1} \otimes B^{-1},\) where \(\otimes \) is denote the Kronecker product and vec denote the vec operator. </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5</span>
    <span class="theorem_thmtitle"><span class="cite">
	[
	<a href="index.html#Sharma 2022iv" >7</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p>  If matrix C is invertible, then the following assertions are equivalent: </p>
<p><span class="rm">(i)</span> the NGAVE <a href="index.html#equ2" class="eqref">2</a> has exactly one solution for any f; </p>
<p><span class="rm">(ii)</span> \(\{ -I + AC^{-1},I + AC^{-1}\} \) holds the column \(\mathcal{W}\)-property; </p>
<p><span class="rm">(iii)</span> \((-I + AC^{-1})\) is invertible and \(\{ I,(-I+AC^{-1})^{-1}(I+AC^{-1})\} \) holds the column \(\mathcal{W}\)-property; </p>
<p><span class="rm">(iv)</span> \((-I+AC^{-1})\) is invertible and \((-I+AC^{-1})^{-1}(I+AC^{-1})\) is a P-matrix; </p>
<p><span class="rm">(v)</span> \((AC^{-1}+(I-2\hat{D}))\) is invertible for any \(\hat{D}\) ; </p>
<p><span class="rm">(vi)</span> \(\{ (-I+AC^{-1})F_1+(I+AC^{-1})F_2\} \) is invertible, where \(F_1,F_2\) \(\in \) \(\mathbb {R}^{n \times n}\) are two arbitrary non-negative diagonal matrices with \(\operatorname {diag}(F_1+F_2) {\gt} 0.\) </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">6</span>
    <span class="theorem_thmtitle"><span class="cite">
	[
	<a href="index.html#Sharma 2022iv" >7</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p>  If matrix C is invertible, then the following assertions are equivalent: </p>
<p><span class="rm">(i)</span> the NGAVE <a href="index.html#equ2" class="eqref">2</a> has a unique solution; </p>
<p><span class="rm">(ii)</span> \(\{ I+AC^{-1},-I+AC^{-1}\} \) has the row \(\mathcal{W}\)-property; </p>
<p><span class="rm">(iii)</span> \((I+AC^{-1})\) is invertible and \(\{ I,(-I+AC^{-1})(I+AC^{-1})^{-1}\} \) has the row \(\mathcal{W}\)-property; </p>
<p><span class="rm">(iv)</span> \((I+AC^{-1})\) is invertible and \((-I+AC^{-1})(I+AC^{-1})^{-1}\) is a P-matrix; </p>
<p><span class="rm">(v)</span> \(\{ F_1(I+AC^{-1})+F_2(-I+AC^{-1})\} \) is invertible, where \(F_1,F_2\) \(\in \) \(\mathbb {R}^{n \times n}\) are two arbitrary non-negative diagonal matrices with \(\operatorname {diag}(F_1+F_2) {\gt} 0.\) </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm3">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">7</span>
    <span class="theorem_thmtitle"><span class="cite">
	[
	<a href="index.html#Sharma 2022iv" >7</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let all diagonal entry of \(AC^{-1}+I\) have the same sign as the corresponding entries of \(AC^{-1}-I\). Then the NGAVE <a href="index.html#equ2" class="eqref">2</a> has exactly one solution for any f if any one of the following conditions is true: </p>
<p><span class="rm">(i)</span> \(AC^{-1}-I\) and \(AC^{-1}+I\) are strictly diagonally dominant by columns; </p>
<p><span class="rm">(ii)</span> \(AC^{-1}-I\), \(AC^{-1}+I\) and all their column representative matrices are irreducibly diagonally dominant by columns. </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm4">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">8</span>
    <span class="theorem_thmtitle"><span class="cite">
	[
	<a href="index.html#Sharma 2022iv" >7</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p> If matrix \(C\) is non-singular, then the NGAVE <a href="index.html#equ2" class="eqref">2</a> has exactly one solution for any f, if the interval matrix \([AC^{-1}-I, AC^{-1}+I]\) is regular. </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm5">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">9</span>
    <span class="theorem_thmtitle"><span class="cite">
	[
	<a href="index.html#Sharma 2022iv" >7</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p>  Let matrix C is non-singular, then the NGAVE <a href="index.html#equ2" class="eqref">2</a> has unique solution for any f if \(\sigma _{\min }(AC^{-1}){\gt}1.\) </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm6">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">10</span>
    <span class="theorem_thmtitle"><span class="cite">
	[
	<a href="index.html#Sharma 2022iv" >7</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p> The NGAVE <a href="index.html#equ2" class="eqref">2</a> has exactly one solution if and only if \(det(A+C)\ne 0\) and for any \(\hat{D}\), matrix \(A-C+2\hat{D}C\) is non-singular. </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm7">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">11</span>
    <span class="theorem_thmtitle"><span class="cite">
	[
	<a href="index.html#Shi Liang Wu 2021" >19</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let matrix A is invertible. The NGAVE <a href="index.html#equ2" class="eqref">2</a> has exactly one solution if \(\rho ((I -2\hat{D})CA^{-1}){\lt}1,\) for any diagonal matrix \(\hat{D}.\) </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm8">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">12</span>
    <span class="theorem_thmtitle"><span class="cite">
	[
	<a href="index.html#Li 2022" >8</a>
	]
</span></span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(\sigma _{\max }(C)\sigma _{\max }(D) {\lt} \sigma _{\min }(A)\sigma _{\min }(B),\) then the new Sylvester-like AVME <a href="index.html#equ1" class="eqref">1</a> has exactly one solution. </p>

  </div>
</div> </p>
<h1 id="section 3">3 Main Results</h1>

<p>This section provides some unique solvability conditions for the new Sylvester-like AVME. First, we see the following result for the NGAVE (<a href="index.html#equ2">2</a>). </p>
<p><div class="theorem_thmwrapper " id="Thm Main 0">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">13</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let matrix A is non-singular. The NGAVE <a href="index.html#equ2" class="eqref">2</a> has exactly one solution if \(\rho (\vert C \vert \cdot \vert A^{-1}\vert ){\lt}1.\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000008">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> For any diagonal matrix \(\hat{D}\), we have \((I -2\hat{D})CA^{-1} \leq \vert (I -2\hat{D})CA^{-1} \vert \leq \vert I -2\hat{D} \vert \cdot \vert CA^{-1} \vert \leq \vert CA^{-1} \vert \leq \vert C \vert \cdot \vert A^{-1} \vert .\) Since \((I -2\hat{D})CA^{-1} \leq \vert C \vert \cdot \vert A^{-1} \vert \), this implies that \(\rho ((I -2\hat{D})CA^{-1}) \leq \rho (\vert C \vert \cdot \vert A^{-1} \vert ) {\lt} 1.\) </p>
<p>Hence, based on <a href="index.html#Thm7">theorem 11</a>, \(\rho (\vert C \vert \cdot \vert A^{-1}\vert ){\lt}1\) implies the unique solvabilty of the NGAVE (<a href="index.html#equ2">2</a>). <div class="proof_wrapper" id="a0000000009">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>For presenting some unique solvability conditions for the new Sylvester-like AVME (<a href="index.html#equ1">1</a>), we first write Eq. (<a href="index.html#equ1">1</a>) into the equivalent NGAVE form (<a href="index.html#equ2">2</a>), and use the results of the NGAVE. </p>
<p>So by taking \(S = B^{T} \otimes A\), \(T = D^{T} \otimes C\), \(f = vec(F)\) and \(x = vec(X)\), where ‘vec’ is vec operator and \( ` \otimes \)’ is the Kronecker product. Then, the new Sylvester-like AVME (<a href="index.html#equ1">1</a>) can be written as the following NGAVE form </p>
<div class="equation" id="equ Equ NGAVE">
<p>
  <div class="equation_content">
    \begin{equation}  \label{equ Equ NGAVE} Sx-|Tx|=f. \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p>Now see the following results for the new Sylvester-like AVME. </p>
<p><div class="theorem_thmwrapper " id="Thm Main 5">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">14</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let C, D be non-singular matrices. The new Sylvester-like AVME <a href="index.html#equ1" class="eqref">1</a> has exactly one solution if \(\sigma _{\min }(D^{-1}B) \sigma _{\min }(AC^{-1}){\gt}1.\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000010">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> To prove the above Theorem, we use Eq. (<a href="index.html#equ Equ NGAVE">6</a>) and <a href="index.html#Thm5">theorem 9</a>. If \(\sigma _{\min }(ST^{-1}) {\gt} 1\), then the Sylvester-like AVME (<a href="index.html#equ1">1</a>) has unique solution. </p>
<p>Now, \(\sigma _{\min }(ST^{-1}) = \sigma _{\min }((B^{T} \otimes A) (D^{T} \otimes C)^{-1}) = \sigma _{\min }((B^{T} \otimes A) (D^{-T} \otimes C^{-1})) = \sigma _{\min }(B^{T} D^{-T} \otimes A C^{-1}) = \sigma _{\min }((D^{-1} B)^{T} \otimes A C^{-1}) = \sigma _{\min }((D^{-1} B) \otimes A C^{-1}) = \sigma _{\min }(D^{-1} B) \sigma _{\min } (A C^{-1}) {\gt} 1.\) </p>
<p>This completes the proof. <div class="proof_wrapper" id="a0000000011">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000012">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">15</span>
  </div>
  <div class="remark_thmcontent">
  <p>In some instances, our result performs better compared to the condition of <a href="index.html#Thm8">theorem 12</a>; see example in <a href="index.html#section 4">section 4</a>. </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm Main 6">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">16</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(S=B^{T} \otimes A,\) \(T=D^{T} \otimes C.\) The new Sylvester-like AVME <a href="index.html#equ1" class="eqref">1</a> has exactly one solution if and only if \(det(S+T)\ne 0\) and for any \(\hat{D}\), matrix \(S-T+2\hat{D}T\) is non-singular. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000013">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> The proof of the above Theorem directly holds by <a href="index.html#Thm6">theorem 10</a> and Eq. (<a href="index.html#equ Equ NGAVE">6</a>). <div class="proof_wrapper" id="a0000000014">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm Main 7">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">17</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let 0 is not an eigenvalue of the matrices A and B. The new Sylvester-like AVME (<a href="index.html#equ1">1</a>) has exactly one solution if \(\rho (\vert D^{T} \vert \cdot \vert B^{-T}\vert ) \cdot \rho (\vert C \vert \cdot \vert A^{-1}\vert ) {\lt}1.\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000015">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Since \(S^{-1} = B^{-T} \otimes A^{-1},\) then </p>
<p>\( \vert T \vert \cdot \vert S^{-1} \vert = \vert D^{T} \otimes C \vert \cdot \vert B^{-T} \otimes A^{-1} \vert = \vert D^{T} \vert \otimes \vert C \vert \cdot \vert B^{-T} \vert \otimes \vert A^{-1} \vert = \vert D^{T} \vert \cdot \vert B^{-T} \vert \otimes \vert C \vert \cdot \vert A^{-1} \vert . \) </p>
<p>Based on spectral radius property of the matrix, we have \(\rho ( \vert D^{T} \vert \cdot \vert B^{-T} \vert \otimes \vert C \vert \cdot \vert A^{-1} \vert )\) = \( \rho ( \vert D^{T} \vert \cdot \vert B^{-T} \vert ) \cdot \rho ( \vert C \vert \cdot \vert A^{-1} \vert ). \) </p>
<p>Based on Eq. (<a href="index.html#equ Equ NGAVE">6</a>) and Theorem <a href="index.html#Thm Main 0">13</a>, if \(\rho (\vert D^{T} \vert \cdot \vert B^{-T}\vert ) \cdot \rho (\vert C \vert \cdot \vert A^{-1}\vert ) {\lt}1,\) then the new Sylvester-like AVME (<a href="index.html#equ1">1</a>) has unique solution for any F. <div class="proof_wrapper" id="a0000000016">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>The new Sylvester-like AVME (<a href="index.html#equ1">1</a>) is equivalently written as NGAVE form (<a href="index.html#equ Equ NGAVE">6</a>). So now we use the results of the NGAVE for the new Sylvester-like AVME. </p>
<p>Now, \(ST^{-1} = (B^{T} \otimes A) (D^{T} \otimes C)^{-1} = (B^{T} \otimes A) (D^{-T} \otimes C^{-1}) = (B^{T} D^{-T} \otimes AC^{-1}) = ( D^{-1}B)^{T} \otimes (AC^{-1}).\) </p>
<p>So, based on ?? we have the following results, see ?? respectively. </p>
<p><div class="theorem_thmwrapper " id="Thm Main 1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">18</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(C,\) \(D\) be non-singular matrices, then the following statements are equivalent: </p>
<p><span class="rm">(i)</span> the new Sylvester-like AVME <a href="index.html#equ1" class="eqref">1</a> has exactly one solution; </p>
<p><span class="rm">(ii)</span> \(\{ (D^{-1}B)^{T} \otimes (AC^{-1})-I,(D^{-1}B)^{T} \otimes (AC^{-1})+I\} \) holds the column \(\mathcal{W}\)-property; </p>
<p><span class="rm">(iii)</span> \(((D^{-1}B)^{T} \otimes (AC^{-1})-I)\) is invertible and \(\{ I,((D^{-1}B)^{T} \otimes (AC^{-1})-I)^{-1}((D^{-1}B)^{T} \otimes (AC^{-1})+I)\} \) holds the column \(\mathcal{W}\)-property; </p>
<p><span class="rm">(iv)</span> \(((D^{-1}B)^{T} \otimes (AC^{-1})-I)\) is invertible and \(((D^{-1}B)^{T} \otimes (AC^{-1})-I)^{-1}((D^{-1}B)^{T} \otimes (AC^{-1})+I)\) is a P-matrix; </p>
<p><span class="rm">(v)</span> \(((D^{-1}B)^{T} \otimes (AC^{-1})+(I-2\hat{D}))\) is invertible for any \(\hat{D}\); </p>
<p><span class="rm">(vi)</span> \(\{ ((D^{-1}B)^{T} \otimes (AC^{-1})-I)F_1+((D^{-1}B)^{T} \otimes (AC^{-1})+I)F_2\} \) is invertible, where \(F_1,F_2\) \(\in \) \(\mathbb {R}^{n \times n}\) are two arbitrary non-negative diagonal matrices with \(\operatorname {diag}(F_1+F_2) {\gt} 0.\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000017">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> The proof of the above Theorem is directly held by <a href="index.html#Thm1">theorem 5</a> and Eq. (<a href="index.html#equ Equ NGAVE">6</a>). <div class="proof_wrapper" id="a0000000018">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm Main 2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">19</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(C,\) \(D\) be non-singular matrices, then the following statements are equivalent: </p>
<p><span class="rm">(i)</span> the new Sylvester-like AVME <a href="index.html#equ1" class="eqref">1</a> has exactly one solution; </p>
<p><span class="rm">(ii)</span> \(\{ (D^{-1}B)^{T} \otimes (AC^{-1})+I,(D^{-1}B)^{T} \otimes (AC^{-1})-I\} \) has the row \(\mathcal{W}\)-property; </p>
<p><span class="rm">(iii)</span> \(((D^{-1}B)^{T} \otimes (AC^{-1})+I)\) is invertible and \(\{ I,((D^{-1}B)^{T} \otimes (AC^{-1})-I)((D^{-1}B)^{T} \otimes (AC^{-1})+I)^{-1}\} \) has the row \(\mathcal{W}\)-property; </p>
<p><span class="rm">(iv)</span> \(((D^{-1}B)^{T} \otimes (AC^{-1})+I)\) is invertible and \(((D^{-1}B)^{T} \otimes (AC^{-1})-I)((D^{-1}B)^{T} \otimes (AC^{-1})+I)^{-1}\) is a P-matrix; </p>
<p><span class="rm">(v)</span> \(\{ F_1((D^{-1}B)^{T} \otimes (AC^{-1})+I)+F_2((D^{-1}B)^{T} \otimes (AC^{-1})-I)\} \) is invertible, where \(F_1,F_2\) \(\in \) \(\mathbb {R}^{n \times n}\) are two arbitrary non-negative diagonal matrices with \(\operatorname {diag}(F_1+F_2) {\gt} 0.\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000019">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> With the help of <a href="index.html#Thm2">theorem 6</a> and Eq. (<a href="index.html#equ Equ NGAVE">6</a>), our result will be true. <div class="proof_wrapper" id="a0000000020">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm Main 3">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">20</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let all diagonal entries of the matrix \((D^{-1}B)^{T} \otimes (AC^{-1})+I\) have the same sign as the corresponding entries of the matrix \((D^{-1}B)^{T} \otimes (AC^{-1})-I\). Then the new Sylvester-like AVME <a href="index.html#equ1" class="eqref">1</a> has exactly one solution for any F if any one of the following conditions is true: </p>
<p><span class="rm">(i)</span> \((D^{-1}B)^{T} \otimes (AC^{-1})-I\) and \((D^{-1}B)^{T} \otimes (AC^{-1})+I\) are strictly diagonally dominant by columns; </p>
<p><span class="rm">(ii)</span> \((D^{-1}B)^{T} \otimes (AC^{-1})-I\), \((D^{-1}B)^{T} \otimes (AC^{-1})+I\) and all their column representative matrices are irreducibly diagonally dominant by columns. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000021">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Applying Eq. (<a href="index.html#equ Equ NGAVE">6</a>) directly into <a href="index.html#Thm3">theorem 7</a>, we get our result. <div class="proof_wrapper" id="a0000000022">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="Thm Main 4">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">21</span>
  </div>
  <div class="theorem_thmcontent">
  <p> If matrices \(C\) and \(D\) are non-singular, then the new Sylvester-like AVME <a href="index.html#equ1" class="eqref">1</a> has a unique solution for any F, if the interval matrix \([(D^{-1}B)^{T} \otimes (AC^{-1})-I, (D^{-1}B)^{T} \otimes (AC^{-1})+I]\) is regular. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000023">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> We come to our result with the help of Eq. (<a href="index.html#equ Equ NGAVE">6</a>) and <a href="index.html#Thm3">theorem 7</a>. <div class="proof_wrapper" id="a0000000024">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<h1 id="section 4">4 A Numerical Example</h1>

<p>In support of our result, we are considering a small example here. However, our result is also applicable to a larger problem. Let’s consider the following matrices for the new class of Sylvester-like AVME (<a href="index.html#equ1">1</a>) </p>
<div class="displaymath" id="a0000000025">
  \begin{equation*}  A= \begin{bmatrix}  3 

&  -4 

&  1

\\ 5 

&  4 

&  1 

\\ -3 

&  5 

&  1 

\end{bmatrix} , ~  B= \begin{bmatrix}  -6 

&  4 

&  2

\\ 3 

&  2 

&  4 

\\ -2 

&  -5 

&  7 

\end{bmatrix} ~  C= \begin{bmatrix}  5 

&  -4 

&  1

\\ 2 

&  2 

&  1 

\\ -2 

&  4 

&  1 

\end{bmatrix}\end{equation*}
</div>
<div class="displaymath" id="a0000000026">
  \begin{equation*}  ~  D= \begin{bmatrix}  5 

&  -4 

&  0

\\ 2 

&  3 

&  2 

\\ -1 

&  1 

&  1 

\end{bmatrix}\\ ~  F= \begin{bmatrix}  -385 

&  -138 

&  -56

\\ -104 

&  -88 

&  103 

\\ 114 

&  -274 

&  61 

\end{bmatrix}\end{equation*}
</div>
<p>It is clear that the condition of Theorem (<a href="index.html#Thm Main 5">14</a>) hold. </p>
<p>\(\sigma _{\min }(D^{-1}B) \sigma _{\min }(AC^{-1}) = 1.1224 \times 0.9154 = 1.027445 {\gt} 1.\) </p>
<p>But condition of the <a href="index.html#Thm8">theorem 12</a> of <span class="cite">
	[
	<a href="index.html#Li 2022" >8</a>
	]
</span> is not satisfying here, since \(\sigma _{\max }(C)\cdot \) \(\sigma _{\max }(D)\) = 7.6562 \(\times \) 6.5791 = 50.3709 \(\nless \) \(\sigma _{\min }(A) \sigma _{\min }(B)\) = 1.3270 \(\times \) 5.0844 = 6.74699. </p>
<p>Moreover, the unique solution of (<a href="index.html#equ1">1</a>) is: </p>
<div class="displaymath" id="a0000000027">
  \begin{equation*}  ~  X= \begin{bmatrix}  4 

&  -3 

&  1

\\ -4 

&  2 

&  2 

\\ 3 

&  -1 

&  5 

\end{bmatrix}. \end{equation*}
</div>
<h1 id="section 5">5 Conclusions</h1>
<p>  In this article, we considered the new class of Sylvester-like AVME \(AXB-|CXD|=F\) and obtained new sufficient results for ensuring the unique solvability of the new class of Sylvester-like AVME (<a href="index.html#equ1">1</a>). We also provided an example in support of our result. Further, the numerical methods for solving the new class of Sylvester-like AVME are also an exciting topic in the future. </p>
<p><div class="acknowledgement_thmwrapper " id="a0000000028">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgements
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The authors express their gratitude to the handling editor and the anonymous referees for their valuable feedback, which greatly enhanced the quality of the manuscript. </p>

  </div>
</div> </p>
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  <dd><p><a href ="https://doi.org/10.1016/0024-3795(93)00184-2"> <i class="sc">R. Sznajder, M.S. Gowda</i>, <i class="it">Generalizations of \(P_{0}\)- and P-properties; Extended vertical and horizontal linear complementarity problems</i>, Linear Algebra Appl., <b class="bf">223</b> (1995), pp. 695–715, <a href="https://doi.org/10.1016/0024-3795(93)00184-2">https://doi.org/10.1016/0024-3795(93)00184-2</a>. </a> </p>
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  <dd><p><a href ="https://doi.org/10.11648/j.ajam.20210904.12"> <i class="sc">K. Xie</i>, <i class="it">On the unique solvability of the generalized absolute value matrix equation</i>, Am. J. Appl. Math., <b class="bf">9</b> (2021) no. 4, p. 104, <a href="https://doi.org/10.11648/j.ajam.20210904.12">https://doi.org/10.11648/j.ajam.20210904.12</a>. </a> </p>
</dd>
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  <dd></dd>
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  <dd> </dd>
</dl>

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