The core inverse and constrained matrix approximation problem

TL;DR

This paper investigates a constrained matrix approximation problem using the core inverse, providing a unique solution, explicit formulas, and new expressions for the core inverse in the context of Frobenius norm minimization.

Contribution

It introduces novel formulas and expressions for the core inverse and solves a constrained matrix approximation problem with a unique solution.

Findings

Derived two Cramer's rules for the unique solution.

Established two new expressions for the core inverse.

Solved the Frobenius norm minimization problem with constraints.

Abstract

In this paper,we study the constrained matrix approximation problem in the Frobenius norm by using the core inverse:\begin{align}\nonumber \left\|{Mx - b} \right\|_F=\min\ \ {\rm subject\ to} \ \ {x\in\mathcal{R}(M)} ,\end{align} where $M\in\mathbb{C}^{\texttt{CM}}_n$. 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.