分块矩阵的初等变换及其应用--毕业论文(3)

洛阳师范学院本科毕业论文

于是有

??I?B?B??n???InI??0???In?BA0?Is????As????AI?s?, ?两边取行列式,得

IInBnIsAI?In?BAIs,

s从而

InBAI?Is?BA, (4) s由(3)、(4)式得

Is?AB?In?BA.

例7 设A?Am?n,B?Bm?n,m?n.证明：?Im?nm?AB???In?AB.[7]证明 构造

???ImA???BI??, n?对其进行如下初等变换

???IA??mr1???Im?AB0?BI???（??A）??r2????, n????BIn??即

??I?A?A???I?n????Im?0In???m?BI????AB0?n?????BIn?, ?或

???ImA?A??r2?(???BI?????1?)?r1????Imn????0I?1?n??BA?, ???I0???IA???IA??n?m????1BIn?????BI???n???m?0I?1BA??,

n???

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洛阳师范学院本科毕业论文

因而

?ImB故

AIn??Im?AB??mIn???1BA,

?Im?AB??m?n?In?BA.

注：此题也可转化为证明矩阵的秩的问题r(?Im?AB)?m?n?r(Im?BA). 3.2分块矩阵在证明矩阵的秩的方面的应用

定理3 设A为m?n矩阵,B为n?l矩阵,若AB?0,则r(A)?r(B)?n.

证明 由于

?A0??Ar(A)?r(B)?r??0B???r??????In?0?r???I?m故

0??0??r??B????In0???n ?0?AB??B??

0??0??r??B????Imr(A)?r(B)?n.

例8 A、B都是n阶方阵,证明:r?AB?A?B??r?A??r?B?. 证明 取分块矩阵

?A0???0B??, ??进行初等变换

?A0?c1?c2?A0?r1?A?r2?A?ABAB?r1?r2?AB?A?B???????????0B?????BB????????B??B?B??????由于分块矩阵不改变它的秩,因此有

AB?B??, ?B??A0??AB?A?Br?A??r?B??r??0B???r??B???AB?B???r?AB?A?B?. B??

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洛阳师范学院本科毕业论文

例9 A、B、C是三个矩阵,证明r?ABC??r?AB??r?BC??r?B?. 证明 取分块矩阵

?0??B?ABC??, 0??对其进行初等变换

?0??B?ABC?0??ABABC??AB??????????B???B?BC??, 0?0?????0??ABr?ABC??r?B??r??B?BC???r?AB??r??BC??r?AB??r?BC?,

??故

r?ABC??r?AB??r?BC??r?B?.

例10 设A、B分别是s?n、n?m矩阵,证明:r(AB)?r(A)?r(B)?n.[8]

证明 只要证

n?r(AB)?r(A)?r(B),

根据

?A0?r??0B???r(A)?r(B), ??有

?Inn?r(AB)?r??0?0??. ?AB?作分块矩阵的初等列变换

?In??0?0?r2?A?r1?In???????AAB???0??In?B???c??????A0?? 2?c1?(?B)AB?????In? ?c?????A2?(?Im)?因此

B??BIn???????c1?c2?0A??, 0????

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洛阳师范学院本科毕业论文

?Inr??0?故

0??BIn????r??r(B)?r(A), ???AB??0A?r(AB)?r(A)?r(B)?n.

定理4 设A为n阶方阵,则A3k?A2k当且仅当r(A2k)?r(Ak?I)?n,

k?1,2,?,n.

证明 令f(?)??2k,g(?)??k?1,则f(?)?(?k?1)?g(?)?1,f(A)?

g(A)?A3k?A2k?0当且仅当r(A2k)?r(Ak?I)?n,即A3k?A2k当且仅当r（A2k）?r(Ak ?E)?n,k?1,2,?,n.

例11 设A为n阶方阵,证明:A2?In?r(I?A)?r(I?A)?n. 证明 构造

0??I?A??, ?0?I?A??对其进行如下初等变换

?I?A??0?0?r2?I?r1?I?A????????I?AI?A??0??I?A??????c2?I?c1??I?AI?A??I?A?? 2I???1??1?（I?A2）0?(I?A2)0?, ?????????2???????21???C1?????(I?A)?2??C2I?A2I02I???????1?r1???(I?A)??r22??因此

1?20???I?A(I?A)0??, ?r??r2?0??I?A???02I??即

r(I?A)?r(I?A)?r(I?A2)?r(I)?r(I?A2)?n,

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洛阳师范学院本科毕业论文

故

r(I?A)?r(I?A)?n?A2?In.

3.3分块矩阵在求解矩阵的逆的方面的应用

????I?A?1. 求一个方阵的逆矩阵可用矩阵的初等变换,即A?I?初等行变换因此.对于分块矩阵求逆.也可以采用分块矩阵的初等变换求解.

?AB?例12 设分块矩阵M???CD??.其中A、B、C、D分别为m?m、m?n、

??[9]

n?m、 n?n矩阵.若A与D?CA?1B都可逆.求M?1.

解 由于

B?AB?r2?CA?1r1?A?(D?CA?1B)?1r2?? M????????? ?CD??0D?CA?1B??????????I0?AB?r1?Br2 ??? ?0I?(D?CA?1B)?1CA?1(D?CA?1B)?1???????A0I?B(D?CA?1B)?1CA?1?B(D?CA?1B)?1?A?1r1??? ??? ?1?1?1?1?1??0I?(D?CAB)CA(D?CAB)???I0A?1?A?1B(D?CA?1B)?1CA?1?A?1B(D?CA?1B)?1??. ??1?1?1?1?1?0I??(D?CAB)CA(D?CAB)??所以 M?1A?1?A?1B(D?CA?1B)?1CA?1?A?1B(D?CA?1B)?1. ??1?1?1?1?1?(D?CAB)CA(D?CAB)特别地.当B?0、C?0时.则

M当B?0、C?0时.则

?1?A?100D?1;

M?1?A?1?A?1BD?10D?1;

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