Analogs of Cramer’s rule for the minimum norm least squares solutions of some matrix equations

Abstract

The least squares solutions with the minimum norm of the matrix equations $\mathbf{AX}=\mathbf{B}\text{,}\mathbf{XA}=\mathbf{B}$ and $\mathbf{AXB}=\mathbf{D}$ are considered in this paper. We use the determinantal representations of the Moore–Penrose inverse obtained earlier by the author and get analogs of the Cramer rule for the minimum norm least squares solutions of these matrix equations.

Introduction

In this paper we shall adopt the following notation. Let ${\mathbb{C}}^{m\times n}$ be the set of m by n matrices with complex entries, ${\mathbb{C}}_r^{m\times n}$be a subset of ${\mathbb{C}}^{m\times n}$ in which any matrix has rank r, ${\mathbf{I}}_m$ be the identity matrix of order m, and $\Vert ·\Vert$ be the Frobenius norm of a matrix.

Denote by ${\mathbf{a}}_{.j}$ and ${\mathbf{a}}_{i.}$ the jth column and the ith row of $\mathbf{A}\in {\mathbb{C}}^{m\times n}$, respectively. Then ${\mathbf{a}}_{.j}^{\ast }$ and ${\mathbf{a}}_{i.}^{\ast }$ denote the jth column and the ith row of a conjugate and transpose matrix ${\mathbf{A}}^{\ast }$ as well. Let ${\mathbf{A}}_{.j}(\mathbf{b})$ denote the matrix obtained from $\mathbf{A}$ by replacing its jth column with the vector $\mathbf{b}$, and by ${\mathbf{A}}_{i.}(\mathbf{b})$ denote the matrix obtained from $\mathbf{A}$ by replacing its ith row with $\mathbf{b}$.

Let $\alpha ≔\{{\alpha }_1\text{,}\dots \text{,}{\alpha }_k\}\subseteq \{1\text{,}\dots \text{,}m\}$ and $\beta ≔\{{\beta }_1\text{,}\dots \text{,}{\beta }_k\}\subseteq \{1\text{,}\dots \text{,}n\}$ be subsets of the order $1⩽k⩽\min \{m\text{,}n\}$. Then $\left|{\mathbf{A}}_{\beta }^{\alpha }\right|$ denotes the minor of $\mathbf{A}$ determined by the rows indexed by $\alpha$ and the columns indexed by $\beta$. Clearly, $\left|{\mathbf{A}}_{\alpha }^{\alpha }\right|$ is a principal minor determined by the rows and columns indexed by $\alpha$. For $1⩽k⩽n$, denote by

$$L_{k\text{,}n}≔\{\alpha :\alpha =({\alpha }_1\text{,}\dots \text{,}{\alpha }_k)\text{,}1⩽{\alpha }_1⩽\cdots ⩽{\alpha }_k⩽n\}$$

the collection of strictly increasing sequences of k integers chosen from the set $\{1\text{,}\dots \text{,}n\}$. For fixed $i\in \alpha$ and $j\in \beta$, let

$$I_{r\text{,}m}\{i\}≔\{\alpha :\alpha \in L_{r\text{,}m}\text{,}i\in \alpha \}\text{,}{J}_{r\text{,}n}\{j\}≔\{\beta :\beta \in L_{r\text{,}n}\text{,}j\in \beta \}\text{.}$$

Matrix equation is one of the important study fields of linear algebra. Linear matrix equations, such as

$$\mathbf{AX}=\mathbf{B}\text{,}$$

$$\mathbf{XA}=\mathbf{B}$$

and

$$\mathbf{AXB}=\mathbf{D}$$

play an important role in linear system theory therefore a large number of papers have presented several methods for solving these matrix equations (see, e.g. [1], [2], [3], [4], [5]). In [6], Khatri and Mitra studied the Hermitian solutions to the matrix Eqs. (1), (3) over the complex field and the system of the Eqs. (1), (2). Wang, in [7], [8], and Li and Wu, in [9] studied the bisymmetric, symmetric and skew-antisymmetric least squares solution to this system over the quaternion skew field. Extreme ranks of matrices in least squares solution of the Eq. (3) over the complex field was investigated in [10] and over the quaternion skew field in [11].

As we know, the Cramer rule gives an explicit expression for the solution of nonsingular linear equations. In [12], Robinson gave its elegant proof over the complex field which aroused great interest in finding determinantal formulas as analogs of the Cramer rule for the matrix equations (see, e.g. [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]). The Cramer rule for solutions of the restricted matrix Eqs. (1), (2), (3) was established in [21].

In this paper, we use the results of [16] to obtain the Cramer rule for least squares solutions of the matrix Eqs. (1), (2), (3) without any restriction. The paper is organized as follows. We start with some basic concepts and results about determinantal representations of the Moore–Penrose inverse in Section 2. In Section 3, we derive some generalized Cramer rules for the minimum norm least squares solutions of the matrix Eqs. (1), (2), (3). In Section 4, we show a numerical example to illustrate the main result.