The core inverse and constrained matrix approximation problem
Hongxing Wang, Xiaoyan Zhang
Abstract
Abstract In this article, we study the constrained matrix approximation problem in the Frobenius norm by using the core inverse: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mo>|</m:mo> <m:mo>|</m:mo> <m:mi>M</m:mi> <m:mi>x</m:mi> <m:mo>−</m:mo> <m:mi>b</m:mi> <m:mo>|</m:mo> <m:msub> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mrow> <m:mi>F</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mspace width=".25em"/> <m:mi>min</m:mi> <m:mspace width="1em"/> <m:mtext>subject</m:mtext> <m:mspace width=".25em"/> <m:mtext>to</m:mtext> <m:mspace width="1em"/> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi>ℛ</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>,</m:mo> </m:math> ||Mx-b|{|}_{F}=\hspace{.25em}\min \hspace{1em}\text{subject}\hspace{.25em}\text{to}\hspace{1em}x\in {\mathcal R} (M), where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>M</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mi>ℂ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mtext>CM</m:mtext> </m:mrow> </m:msubsup> </m:math> M\in {{\mathbb{C}}}_{n}^{\text{CM}} . We get the unique solution to the problem, provide two Cramer’s rules for the unique solution and establish two new expressions for the core inverse.