[A,B]行块矩阵penrose型广义逆的充要条件

The Necessary and Sufficient Conditions for Penrose—inverses of the Matrix[A,B]

设A∈C^(m×n),B∈C^(m×p)及四个矩阵方程:1)AGA=A,2)GAG=G,3)(AG)~*=AG,4)(GA)~*=GA如果G满足上述方程i),j),…k),则称G为(ij…k)型逆或penrose型广义逆,简称广义逆,并记为A(ij…k).其全体记为A{ij…k},利用矩阵广义逆的理论研究了下列两类等式成立的的充要条件:I)其中α+β=1,α>0,β>0,1≤i<j≤4.其研究结果推广了李小彬等人的结论,因而也是2007年田永革教授在国际线性代数学会会刊上所获得的相应结果的进一步推广.

Let A∈C^m×n,B∈C^m×p and four equations：1）AGA=A,2）GAG=G,3）（AG）＊=AG,4）（GA）＊=GAIf G satisfies equation i）, j）,.- ~ , k） etc, then G is called an （ij... k）-inverse or penrose-inverse of A and is denoted as A（ijk）. A{ij...k} denotes the set of （ij...k）-inverses. In this paper,we study the following two kinds of eqalities：where α＋β=1,α〉0,β〉0,1≤i〈j≤4 .The necessary and sufficient conditions about them are derived by using generalized-inverse theory , they generalized the previous work of Li. （2008）and Tian.（2007）.