On Real Matrices to Least-Squares g-Inverse and Minimum Norm g-Inverse of Quaternion Matrices *

Through the real representations of quaternion matrices and matrix rank method, we give the expression of the real matrices in least-squares g-inverse and minimum norm g-inverse. From these formulas, we derive the extreme ranks of the real matrices. As applications, we establish necessary and sufficient conditions for some special least-squares g-inverse and minimum norm g-inverse.


Introduction
Throughout this paper, stands for the real number field, stands for the set of all m matrices over the quaternion algebra , , , .
a a i a j a k i j k ijk a a a a I, A T , A * and † A stand for the identity matrix' the transpose' the conjugate transpose and the Moore-Penrose inverse of a quaternion matrix A. In [1], for a quaternion matrix A, is called the rank of a quaternion matrix A and denoted by  .  4) XA XA  A matrix X is called a least-squares g-inverse of A if it satisfies both 1) and 3) in the Penrose equations, and denoted by a matrix X is called a minimum norm g-inverse of A if it satisfies both 1) and 4) in the Penrose equations, and denoted by The general expression of   1,3 ; A can be written as  
For an arbitrary quaternion matrix we define a map ( ) By (5), it is easy to verify that ( ) The least-squares g-inverse and minimum norm g-inverse have a very wide range of applications in numerical analysis and mathematical statistics and have been examined by many authors(see, e.g., [3][4][5][6]).Haruo [3] developed some equivalent conditions on least-squares general inverse in 1990.Tian [4] presented the maximal and minimal ranks of the Schur complement to least-squares g-inverse and minimum norm g-inverse in 2004.Tian [5] establish necessary and sufficient conditions for a matrix to be the least-squares g-inverse and minimum norm g-inverse from rank formulas in 2005.Guo, Wei and Wang [6] derived structures of least squares g-inverses and minimum norm g-inverse of a bordered matrix in 2006.
Noticing that the properties of the real matrices in least-squares general inverse   1,3

A
and minimum norm g-inverse   1,4

A
(4) have not been considered so far in the literature.We in this paper use the real representations of quaternion matrices and matrix rank method to investigate (4) over .In Section 2, we first give the expression of the real matrices i and in (4), then determine the maximal and minimal ranks of the real matrices i and i in (4).As applications, we establish necessary and sufficient conditions for a quaternion matrix has a pure real or pure imaginary A and 1,4 A .The necessary and sufficient conditions for all 1,  A are pure real or pure imaginary of a quaternion matrix are also presented.

Main Results
We begin with the following lemmas which proof just like those over the complex field.
ij i j

X P XQ
From (1), the least-squares g-inverse of   A  can be written as where and   , , , Substituting them into (8) yields the four real matrices B 0 , B 1 , B 2 and B 3 in ( 9)- (12).
According to Lemma 2 and Theorem 2, we can get the following extreme ranks formulas for the real matrices in the least-squares g-inverses.
Theorem 2.4 Suppose that A and   1,3 A are defined as (3) and (4).Then Proof.Applying ( 6) and ( 7) to B 0 in (9), we get the following By Lemma 1, it is not difficult to find that , , , A A A A and  A are defined as above.By the same manner, we can get extreme ranks of B 1 , B 2 and B 3 .
As one of important applications of the maximal and minimal ranks to real matrices, Theorem 2 can help to get the necessary and sufficient conditions for the existence of some special least-squares g-inverses.We show them in the following.

b) All least-squares g-inverses of quaternion matrix A are real matrices if and only if
where 1 2 3 , , A A A and  A are defined as Theorem 2. Corollary 2.6 Suppose
Theorem 2.8 Suppose that A and   1,4 A are defined as (3) and (4).Then As one of important applications of the maximal and minimal ranks to real matrices, Theorem 4 can help to get the necessary and sufficient conditions for the existence of some special minimum norm g-inverse.We show them in the following.
Corollary 2.9 Suppose , m where 1 2 3 , , A A A and   A are defined as Theorem 4. Corollary 2.10 Suppose minimum norm g-inverse of  ,

1 ,
W 2 , W 3 and W 4 are arbitrary real matrices with compatible sizes.
quaternion matrix A has a real minimum norm g-inverse if and only if All minimum norm g-inverse of quaternion matrix A are real matrices if and only if quaternion matrix A has a pure imaginary minimum norm g-inverse if and only if All minimum norm g-inverse of quaternion matrix A are pure imaginary matrices if and only if and  A are defined as Theorem 4.