[][][][][]2**11**2===..........................mn mr rn u x x x x v r r y x xx w x x xy uvw xx r r xx S A B S m A m S n B n A r B r f z u v w f z u v w f u u z z f f f v v w w =⎡⎤⎡⎤⎡⎤⎢⎥==⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎢⎥⎣⎦⎡⎤⎢⎥⎡⎤==⎣⎦⎢⎥⎢⎥⎣⎦，其中的行数的行数，的列数的列数，的列数的行数另外矩阵的元素一般是同一类型,于是有1.25785475855273849563421021405251203241410530330435350233343021544104415054213042111AB ⨯+⨯+⨯⨯+⨯+⨯⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⨯+⨯+⨯⨯+⨯+⨯⎢⎥⎢⎥⎢⎥⎢⎥===⎢⎥⎢⎥⎢⎥⎢⎥⨯+⨯+⨯⨯+⨯+⨯⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⨯+⨯+⨯⨯+⨯+⨯⎣⎦⎣⎦⎣⎦1112223331.3210121312222421134435()()6862342x y z X x Y y Z z x y z A B Y AX Z BYZ BY B AX BA X ⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥===⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦-⎡⎤⎡⎤⎢⎥⎢⎥=-=-⎢⎥⎢⎥⎢⎥⎢⎥---⎣⎦⎣⎦==-⎡⎢====----⎣记，，一般变量都记作列向量，这个称为系数矩阵，即系数按照原来的位置排成矩阵于是有，线性变换的矩阵写法从而11112322212333312343543568668623422342Xz x z x x x z x z x x x z x z x x x ⎤⎥⎢⎥⎢⎥⎦-=-+⎡⎤⎡⎤⎡⎤⎧⎪⎢⎥⎢⎥⎢⎥⇒=-⇔=+-⎨⎢⎥⎢⎥⎢⎥⎪⎢⎥⎢⎥⎢⎥---=---⎣⎦⎣⎦⎣⎦⎩线性变换的方程组写法和矩阵写法123111111(1)323124111211105111111111213111213(1)112(1)31311214111214(1)112(1)4101513101513(1)015(1)3BA A ⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥-=-----⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥--⎣⎦⎣⎦⎣⎦⨯+⨯+⨯⨯+⨯+⨯-⨯+⨯-+⨯=-⨯-⨯+⨯-⨯-⨯+⨯--⨯-⨯-+⨯⨯+⨯+⨯⨯+⨯+⨯-⨯+⨯-+⨯1112111111160211136303221212131752111313(7)3521212(1)64411136343(4)212(1)21180622232115181212⎡⎤⎡⎤⎢⎥⎢⎥--⎢⎥⎢⎥⎢⎥⎢⎥-⎣⎦⎣⎦⨯⨯⨯⨯⨯⨯⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥=---=⨯⨯-⨯-⨯⨯⨯-⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥--⨯⨯⨯-⨯⨯-⨯⎣⎦⎣⎦⎣⎦⎣⎦⎡⎤⎢⎥=--⎢⎥⎢⎥-⎣⎦182026216242223221215(2)1231722218212(2)1221614142132232217204292123111110(2)124111225051111341TT AB A B A ----⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥-=-----=-⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥-------⎣⎦⎣⎦⎣⎦-⎡⎤⎢⎥-=--⎢⎥⎢⎥-⎣⎦-⎡⎤⎡⎤⎡⎢⎥⎢⎥=---=-⎢⎥⎢⎥⎢⎥⎢⎥-⎣⎦⎣⎦书上是11111111111110111110(1)111(1)0121215121215(1)212(1)5131411131411(1)314(1)11002559860⎤⎡⎤⎢⎥⎢⎥-⎢⎥⎢⎥⎢⎥⎢⎥-⎣⎦⎣⎦⨯-⨯+⨯⨯-⨯+⨯-⨯-⨯-+⨯⎡⎤⎢⎥=⨯-⨯+⨯⨯-⨯+⨯-⨯-⨯-+⨯⎢⎥⎢⎥⨯+⨯+⨯⨯+⨯+⨯-⨯+⨯-+⨯⎣⎦⎡⎤⎢⎥=-⎢⎥⎢⎥⎣⎦111122221122111122221111112222220000(,,,)00,,,nn nn nn nn nn k kk k nn nn a a a a diag a a a a a a b a b A B a b a b a a b a A AB a a ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎡⎤⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎣⎦对角矩阵的三种表示表示形式:.规律:记则11111221122111121212221212,,,,.(1,:)(2,:)(,:)(,:)(,,,(,:)nn nn nn r r i i n n nr b a a a a a A A a c c c C c c c C C C i C i C i c c c c c c C n ----⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎣⎦⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥===⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦当且仅当全不为零时,是可逆矩阵,并有记,其中表示的第行,1111111211111222122222222212111212122212),(1,:)(2,:).(,:)(:,1)(:,2)ir r r ii nn n nn n nn nr nn n n s s sn a c a c a c a C a c a c a c a C AC a C i a c a c a c a C n d d d d d d D D D d d d ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎡⎤⎢⎥⎢⎥==⎢⎥⎢⎥⎣⎦ 则,即分别乘以的第行记[][]1211112211111222222211221122(:,)(:,)(:,),(:,1)(:,2)(:,),.j j sj nn n nn n nn jj snsn nn sn d d D n D j D j D j d a d a d a d a d a d a d DA a D a D a D n a D j a d a d a d ⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎣⎦⎡⎤⎢⎥⎢⎥==⎢⎥⎢⎥⎣⎦ ,其中表示的第列,则即分别乘以的第列[]111213212223313233(1),10010,001(1,:)(1,:)(2,:),(:,2)(:,1)(:,2)(:,3),(3,:)(1,:),(2,:),(3,:)1,2,3(:,1AB BA a a a A a a a C k C a a a A CA kA A AC kA A A A A A A A A A ⎡⎤⎡⎤⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎡⎤⎢⎥=+=+⎢⎥⎢⎥⎣⎦由1.5规律可得.，是一个初等矩阵则其中表示的第行,()1112112122221212111112121121212222221122),(:,2),(:,3)1,2,3(2)n n n n n nn n n n n n n n n n nn n n A A A a a a x aa a x x x x a a a x a x x a x x a x x a x x a x x a x x a x x a x x a x x ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦=++++++++++++ 表示的第列,这是初等矩阵的性质,第3章.规律:这是二次型,第5章.22221.8(2)(1),,,()()()()AB A BA B AB A AB A AAA A BA A AB A BA B BA B BBB B AB B BA B ===⇒=⇒=⇒==⇒=⇒=⇒=由于是两边同时右乘矩阵结合律两边同时右乘矩阵结合律222221.91(1)(1)(2)2()21(2)(1)(2)2()2X Y A X A BA X A BA X Y BA Y A BA Y A BA ⎧+=+⇒=+⇒=+⎪⎪⎨⎪-=-⇒=-⇒=-⎪⎩222222222222222(1)()()(),2(2)()(),(3)0()0111100000,111100A B A B A B A BA AB B AB BA A BA AB B A AB B A B A B A BA AB B AB BA A BA AB B A B A A A A A A E AB A B +=++=+++≠+++≠+++-=+--≠+-+≠-=⇒-=⇒-=-⎛⎫⎛⎫⎛⎫==== ⎪⎪ ⎪---⎝⎭⎝⎭⎝⎭因为所以因为所以因为不能推出必有或者例如2211()0000100(4)00,000011100100(5),0001001010,()()A A E A A E A A Ax Ay x y A Ax Ay A Ax A Ay x y---==-=⎛⎫⎛⎫=== ⎪ ⎪⎝⎭⎝⎭⎛⎫⎛⎫⎛⎫⎛⎫ ⎪ ⎪=== ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭ ⎪ ⎪⎝⎭⎝⎭=⇒=⇒=所以不能推出必有或者不能推出必有例如不能推出必有例如若可逆则11111(1),0. 2.31202211111,,1011010222122123131()100121224(2)a b d b A ad bc A c d c a ad bc A A B B AB AB -----⎛⎫⎛⎫=-≠= ⎪ ⎪--⎝⎭⎝⎭--⎛⎫⎛⎫⎛⎫⎛⎫=⇒==⇒= ⎪ ⎪ ⎪ ⎪-⎝⎭⎝⎭⎝⎭⎝⎭--⎛⎫⎛⎫⎛⎫⎛⎫==⇒= ⎪⎪ ⎪ ⎪---⎝⎭⎝⎭⎝⎭⎝⎭已知当时,有见节设准对角矩阵(一种特殊的分块1122121212111111221220000,,00000,,,,,,,,,,,s s s s s k k k k s s s s A A A A A A A B B A A A A B B B B B B A B A A A B A A AB A A A B A A ----⎛⎫⎛⎫ ⎪ ⎪⎪ ⎪= ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎛⎫ ⎪ ⎪= ⎪ ⎪⎝⎭⎛⎫⎛⎫⎪ ⎪ ⎪⎪=== ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭ 矩阵)即其中都为方阵,准对角矩阵其中都为方阵.于是有:1111.010*********,.011000220001A A C C B B ---⎛⎫ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭-⎛⎫ ⎪ ⎪⎛⎫⎛⎫ ⎪=== ⎪ ⎪ ⎪⎝⎭⎝⎭- ⎪⎪ ⎪⎝⎭因此,就有21222221.1240()4040,11(1)01124(1)42122n A A A E A E A E X E A E A E X A A E x x A E x x x a b x a x a b a a x x x a x a b a b b x x -+-=----=+-=⇒+-=-⇒--+=+---=-==⎧⎧+-≡+---⇒⇒⎨⎨--=-=⎩⎩-+=⇒设阶方阵，且，证可逆。