线性代数第一章

1、103020345122 44.54.5422.62.42.25.86.266 11121314212223243132333441424344aaaaaaaaaaaaaaaa 0111100001001010 11yx032321.52.522.5311.533112 a0234121228A 1231231232344232282xxxxxxxxx 423432121228B 123xxxx 432b 111212122212nnmmmnaaaaaaaaa()ijm nAa ()ijm nBb ,1,1ijijabimjn 矩阵是一个矩阵是一个矩形阵列矩形阵列.它是高等代数学中的常见它是高

2、等代数学中的常见工具，也常见于统计分析等应用数学学科中工具，也常见于统计分析等应用数学学科中.矩矩阵的运算是数值分析领域的重要问题阵的运算是数值分析领域的重要问题.12naaa 12naaa 111212122212nnnnnnnn naaaaaaAaaa 21000000n 12,ndiag , , k kdiagk000000111nnE 3003 200020002 11121222000nnnnn naaaaaa 11212212000nnnnn naaaaaa 1111naa 01111naa 01332 02021011x 0330 020201010 0000 000 00000

3、0 111212122212nnmmmnaaaaaakkkkkkkkkAAaakka 111112121121212222221122nnnnmmmmmnmnababababababABababab 矩阵加法的前提：两个矩阵必须矩阵加法的前提：两个矩阵必须同型同型. 设设A,B是同型矩阵，则它们的是同型矩阵，则它们的差差定义为定义为A+(B),记为记为AB，即即AB =A+(B). 矩阵的数乘和加法运算合称为矩阵的矩阵的数乘和加法运算合称为矩阵的线性运算线性运算.918815,077902AB 1(3)2XAB 191912292119222 44.54.5422.62.42.25.86.26

4、6 103020345122 11212sijijijs ij sjkkk ica ba ba ba b 212221121331aaaaaa 112131122232bbbbbb 112131111213bbaaa b 122232111213bbaaa b 112131212223bbaaa b 122232212223bbaaa b 11212sijijijs ij sjkkk ica ba ba ba b 只有当矩阵只有当矩阵A 的的列数列数等于矩阵等于矩阵B 的的行数行数时，乘积时，乘积AB才有意义才有意义. 否则，不能相乘否则，不能相乘. 乘积矩阵乘积矩阵C 的行数与矩阵的行数与矩

5、阵A 的行数相同，的行数相同， 矩阵矩阵C 的列数与矩阵的列数与矩阵B 的列数相同的列数相同41010311132102201134AB 与与41010311132102201134AB 9219911 24241236AB 与与24241236AB 24243612BA 1632816 0000 矩阵的乘法矩阵的乘法不满足交换律不满足交换律. AB是是A左乘左乘B，BA是是A右乘右乘B. 如果两个如果两个n阶方阵阶方阵A,B，若，若AB=BA，称，称A与与B是是可可交换的交换的.010210,000000ABC , ,0000AB 0000AC ,ABACAO BC 矩阵的乘法矩阵的乘法不满

6、足消去律不满足消去律. 矩阵的乘法矩阵的乘法不满足交换律不满足交换律. 矩阵的乘法矩阵的乘法不满足消去律不满足消去律.22.5311.533112A 11yx032321.52.5a0cossin66sincos66R BRA 0.2330.6652.0980.3660.2993.5982.1162.3361.3662.482 a0图形绕原点旋图形绕原点旋转了转了30度度234121228A 1231231232344232282xxxxxxxxx 123xxxx 432b Axb kAAAA (AB)k = AkBk. (A+B)2 = A2+2AB+B2. Ak=O , A不一定为不一定为

7、O. 对角阵对角阵 12000000n k 12,kkkndiag 12 ,1233BC ABC 123246369 12 ,1233BC 123246369ABC 100100()ABC ()()()()BCBCBCBC ()()()B CB CBCB C 9914 BC 112323CB 9912314246369 14 11121314212223243132333441424344aaaaaaaaaaaaaaaa 0111100001001010 A 0111100001001010A 2A 2110011110000211 111212122122mnnmnmmnaaaaaAaaaa

8、 122111121222TnmmmnmnnAaaaaaaaaa A是对称矩阵是对称矩阵 AT = A. A是反对称矩阵是反对称矩阵 AT = A. A是方阵是方阵 (A+AT)T = A+AT(A AT)T = (A AT)1201A 410()04f A 0012456710121000100010058A 23EBCO 112112112222nmnmnmmnaaaaaAaaaa 11211222212211,nmmnnmnaaAaaAaaaAaa 12nm nAAAA 111212122122mnnmnmmnaaaaaAaaaa 22121111211222,nmmnnmnAAAaaa

9、aaaaaa 12mAAAA 12m nsOOOOAOOAAA 1000000000002100210120000A 21OAAOA 111212122212rrsssrAAAAAAAAAA 111212122212rrsssrAAAAAAAAAkkkkkkkkkkA 111212122212rrsssrAAAAAAAAAA 111212122212rrsssrBBBBBBBBBB 111112111121212222221122rrrrsssssrsrABABABABABABABABABAB 111212122212ttssstAAAAAAAAAA 111212122212rrtttrBBB

10、BBBBBBB 111212122212rrsssrCCCCCCABCCC 1(1, ;1, )tijikkjkCA Bis jr 10001010010012011210104111011120AB 与与10001010010012011210104111011120AB 1000010012101101A 1010120110411120B 1EOAE 112122BEBB 1111121122BEABA BBAB 1010120124331131 111212122212rrsssrAAAAAAAAAA 112111222212TTTsTTTTsTTTrrsrAAAAAAAAAA 123

11、1231232344232282xxxxxxxxx 12312312323234441xxxxxxxxx 1232323232242xxxxxxx 12323323220 xxxxxx 11231231232344232282xxxxxxxxx 12312312323234441xxxxxxxxx 1232323232242xxxxxxx 12323323220 xxxxxx 234412132282 121323441141 121301220132 121301220010 11230023120001400000 10010020120001600003 1123002312000140

12、0000 10300012000001000001 任何一个矩阵都可以经过有限次任何一个矩阵都可以经过有限次初等行变换初等行变换化为化为行最简矩阵行最简矩阵，且行最简矩阵，且行最简矩阵唯一唯一.123212024231373283023743rrr 231371202464166023743A 231371202464166023743A 2131412321202401111088912077811rrrrrr 123212024231373283023743rrrA 23242( 1)8712024011110001400014rrrrr 4312024011110001400000rr

13、2313212004011030001400000rrrr 12210202011030001400000rr 123100221010343001A 126511131121322B 35221001320103001111 100310134000000( )rrm nEOEOO 1 02020 11 030 00140 0000A 3132515251223410000010000001000000cccccccccc 341 0 0 0 00 1 0 0 00 0 1 0 00 0 0 0 0cc 任何一个矩阵都可以经过有限次任何一个矩阵都可以经过有限次初等行变换初等行变换化为化为行最

14、简矩阵行最简矩阵，再经过有限次初等列变换化成，再经过有限次初等列变换化成等价等价标准型标准型.011111( , )1101E i j 11( ( )11E i kk 11( , ( )11Eki j k A A 111212122212nnnnnnaaaaaaDaaa 1212nppnpaaa共共 n!个个( 1) a12a21a11a2211122122aaaa 111213212223313233aaaaaaaaa a11a22a33 a12a23a31 a13a21a32 a11a23a32 a12a21a33 a13a22a31 对角线对角线法则法则= a11 a22ann .112

15、12212000nnnnaaaaaa11121222000nnnnaaaaaa三角形三角形行列式行列式= a1n a2 n-1an1 .1122000000nnaaa12110000000nnnaaa 对角型对角型行列式行列式(1)2( 1)n n 11121111211112111221212121212nnniiiiininiiiniiinnnnnnnnnnnnnaaaaaaaaabcbcbcbbbcccaaaaaaaaa12311121332112312311112332132112345678131324245768576811121111211122121212nnijijinjni

16、iinnnnnnnnnaaaaaaakaakaakaaaaaaaaaa = a11 a22ann .11121222000nnnnaaaaaa3112513420111533D 3112513420111533D 1312153402115133 c1c2 13120846021101627 r2 r1r4 5r1 r3+2r2r4 8r2131202112042301627 1312021120045001015 r2r3 第二行第二行提出提出2521312021120045000 4352rr 40 1234234134124123D 123221343D 2 160 nxaaaxaDaa

17、x nxaaaxaDaax (1)(1)(1)xnaxnaxnaaxaaax 将将2至至n行都行都加到第加到第1行行 111(1) )axaxnaaax 11100(1) )00 xaxnaxa 1(1) )()nxna xa 223234113D 112143A 122133A 132234A 123 2( 3) 22 1234101231101205D 1234101231101205D 123rr 32rr 2202101241021205 222412125 2 3( 1) 12cc 32cc 020311323 3133 1 2( 1) ( 1) 2 24 1122,0,ijijin

18、jnDija Aa Aa Aij 1122,0,ijijninjDija Aa Aa Aij 2nababDcdcd |22AAEO264AAEE (2)(3)4AEAEE 11(2)(3)4AEAE 22AAEO22AAE ()2A AEE 1( ()2AAEE 11()2AAE 1121112222*12nnnnnnAAAAAAAAAA 1121112222*12nnnnnnAAAAAAAAAA abAcd 11A12Ad c 21Ab 22Aa *dbAca 123221343A 11A12A2 3 13A2 *264365222A 216A 226A 232A 314A 325A 33

19、2A 1121112222*12nnnnnnAAAAAAAAAA 11121112112122212222*1212nnnnnnnnnnnnaaaAAAaaaAAAAAaaaAAA 1*1AAA 1234A 1234A 2 *4231A 1421231A 123221343A 2 123221343A *264365222A 135221323111A 123221343A 123221310001000143EA 135221323111000100110rAE 352211323111A 行最简形行最简形矩阵矩阵12210132 ,0121312AB 174298110 ,1153165A

20、X 12210132 ,0121312AB 122102132 ,113213AC 111213142122232431323334aaaaaaaaaaaa11133133aaaakkmnC C401204180322A 401204180322A 401204180322A 22001 规定规定零矩阵零矩阵的秩为的秩为0. 若矩阵若矩阵A中有一个中有一个s阶的子式不为零，则阶的子式不为零，则r(A) s . 若矩阵若矩阵A中所有中所有t 阶的子式全为零，则阶的子式全为零，则r(A) t . 若若A为为n阶矩阵，则阶矩阵，则A的的n阶子式只有一个阶子式只有一个, 即即|A|当当|A|0 时，时，r(A)= n ,可逆矩阵（非奇异矩阵）又可逆矩阵（非奇异矩阵）