投影阵的构造

THE STRUCTURE OF PROJECTIVE MATRICES

给出了向μ（Ａ）的投影阵Ｐ的５种构造形式不同的集合表示：设Ａ，Ｂ分别为ｎ×ｍ，ｎ×ｐ阶固定矩阵，μ（Ａ）∩μ（Ｂ）＝｛０｝，且μ（Ａ）μ（Ｂ）Ｒｎ，令Ｓ１＝｛Ｐ：Ｐ２＝Ｐ，ＰＡ＝Ａ，ＰＢ＝０，μ（Ｐ）＝μ（Ａ）｝，Ｓ２＝｛（Ａ…０）Ｇ：Ｇ∈｛（Ａ…Ｂ）－｝｝，Ｓ３＝｛ＡＡ′Ｇ：Ｇ∈｛（ＡＡ′＋ＢＢ′）－｝｝，Ｓ４＝｛ＡＡ′（ＡＡ′＋ＢＢ′＋ＣＣ′）－１：Ｃ满足μ（Ａ）μ（Ｂ）μ（Ｃ）＝Ｒｎ｝，Ｓ５＝｛ＡＧＢ⊥′：Ｇ∈｛（Ｂ⊥′Ａ）－｝｝。

This paper gives five different forms of set representations for the projective matrices towards the space μ(A). Assume that A, B are n×m, n×p given matrices respectively, μ(A)∩μ(B)={0}, and μ(A)μ(B)R n. Let S 1={P:P 2=P,PA=A,PB=0,μ(P)=μ(A)}, S 2={(A…0)G:G∈{(A…B) -}},S 3={AA′G:G∈{(AA′+BB′) -}}, S 4={AA′(AA′+BB′+CC′) -1 :C satisfies the formula μ(A)μ(B)μ(C)=R n}, S 5={AGB ⊥′ :G∈{(B ⊥′ A) -}}. Then S 1=S 2=S 3=S 4=S 5.