Abstract
<p> The main goal of the present study is to solve fuzzy linear system\index{fuzzy linear system}s (FLS) represented as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A ModifyingAbove bold x With tilde left-parenthesis rho right-parenthesis equals ModifyingAbove bold y With tilde left-parenthesis rho right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">x</mml:mi> </mml:mrow> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">y</mml:mi> </mml:mrow> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A \widetilde {\mathbf {x}}(\rho )=\widetilde {\mathbf {y}}(\rho )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where the coefficient matrix <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a rectangular complex matrix and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove bold x With tilde left-parenthesis rho right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">x</mml:mi> </mml:mrow> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\widetilde {\mathbf {x}}(\rho )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove bold y With tilde left-parenthesis rho right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">y</mml:mi> </mml:mrow> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\widetilde {\mathbf {y}}(\rho )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are appropriate vectors with fuzzy entries. The standard approach to solving FLS with a crisp coefficient matrix involves solving the equivalent crisp linear system (CLS) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S bold x left-parenthesis rho right-parenthesis equals bold y left-parenthesis rho right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">x</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">y</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">S\mathbf {x}(\rho )=\mathbf {y}(\rho )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where coefficient matrix is given in a suitable block matrix form. Proper block representations of the weighted Moore-Penrose inverse\index{weighted Moore-Penrose inverse} (WMPI) and the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -weighted Drazin inverse\index{ <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -weighted Drazin inverse} (WDI) of the coefficient matrix <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are derived. </p>