The core inverse and constrained matrix approximation problem

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.

Let M ∈ C n×n be singular. The smallest positive integer k for which rk M k+1 = rk M k is called the index of M and is denoted by Ind(M ).
The index of a non-singular matrix is 0 and the index of a null matrix is 1. In particular, when M ∈ C CM n , the matrix X is the group 10 inverse of M , and is denoted by X = M # , (see [4,8,25]).
The core inverse of M ∈ C CM n is defined as the unique matrix X ∈ C n×n satisfying the equations: M XM = M , M X 2 = X and (M X) * = M X, and is denoted by X = M # , (see [1,30]). It is noteworthy that the core inverse is a "least squares" inverse, (see [7,18]). Moreover, it is proved that Recently, the relevant conclusions of the core inverse are very rich. In [2,18,19,28], generalizations of core inverse are introduced, for example, the core-EP inverse and the weak group inverse, etc. In [16,20,27,29], their algebraic properties and calculating methods are studied. In [9,22], the studying of them 20 is extended to some new fields, for example, ring and operator, etc. Moreover, the inverses are used to study partial orders in [1,28,30,31].
Consider the following equation: Let M ∈ C n×n with Ind(M ) = k, and b ∈ R M k . Campbell and Meyer [5] show that x = M D b is the unique solution of (1.2) with respect to x ∈ R M k .
Wei [32] gets the minimal P -norm solution of (1.2), where P is nonsingular, P −1 M P is the Jordan canonical form of M and x p = P −1 x 2 . Furthermore, let M ∈ C m×n . Wei [33] considers the unique solution of More results of (1.2) under some certain conditions can be found in [6,21,23,24,25,32,34]. where M ∈ C CM n , rk(M ) = r < n and b ∈ C n .
where T is nonsingular. Furthermore,

Main Results
is the unique solution of (1.3).

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Proof. From x ∈ R (M ), it follows that there exists y ∈ C n for which x = M y.
Let the decomposition of M be as in (2.2). Denote Since T is invertible, we have min y1,y2 Therefore, When M ∈ C n×n is nonsingular, it is well-known that the solution of (1.2) is unique and is called the Cramer's rule for solving (1.2). In [3], Ben-Israel gets a Cramer's rule for obtaining the least-norm solution of the consistent linear system (1.2), where U are V are of full column rank, R (U ) = N (M * ) and R (V ) = N (M ).
In [24], Wang gives a Cramer's rule for the unique solution x ∈ R M k of (1. 2) can be found in [4,12,13,14,15,25,26]. In the following Theorem 3.4 and Theorem 3.6, we will give two Cramer's rules for the unique solution of (1.3).

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First of all, we give the following two lemmas to prepare for a Cramer's ruler for the core inverse in Theorem 3.4.
where i = 1, 2, . . . , n. In the following Lemma 3.3 and Theorem 3.4, we give the 55 unique least-squares solution of (1.3) in a similar way under weaker conditions. is invertible and that is, G is invertible and G −1 is of the form (3.13).
Based on Lemma 3.2 and Lemma 3.3, we get a Cramer's rule for the unique solution of (1.3).
THEOREM 3.4. Let M and b be as in (2.2), and let L be as in Lemma 3.2.
Proof. Since G is invertible, applying Lemma 3.3, we get the unique solution Applying (3.3) we obtain (3.14).
In the following theorem, we give a characterization of the core inverse and prepare for a Cramer's ruler for the core inverse in Theorem 3.6. It follows that (3.15).
Proof. Applying Theorem 3.5 to Theorem 3.1, we have that is, It follows from (3.3) that we get (3.16).
In [10], Ji obtains the condensed determinantal expressions of M † and M D .
By using Theorem 3.5, we get a condensed determinantal expression of M # .
THEOREM 3.7. Let M and L be defined as in (3.12). Then the core inverse M # is given:

Disclosure statement
No potential conflict of interest was reported by the authors.