線性代數(shù)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 矩陣是一個(gè)矩陣是一個(gè)矩形陣列矩形陣列.它是高等代數(shù)學(xué)中的常見它是高

2、等代數(shù)學(xué)中的常見工具，也常見于統(tǒng)計(jì)分析等應(yīng)用數(shù)學(xué)學(xué)科中工具，也常見于統(tǒng)計(jì)分析等應(yīng)用數(shù)學(xué)學(xué)科中.矩矩陣的運(yùn)算是數(shù)值分析領(lǐng)域的重要問題陣的運(yùn)算是數(shù)值分析領(lǐng)域的重要問題.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 矩陣加法的前提：兩個(gè)矩陣必須矩陣加法的前提：兩個(gè)矩陣必須同型同型. 設(shè)設(shè)A,B是同型矩陣，則它們的是同型矩陣，則它們的差差定義為定義為A+(B),記為記為AB，即即AB =A+(B). 矩陣的數(shù)乘和加法運(yùn)算合稱為矩陣的矩陣的數(shù)乘和加法運(yùn)算合稱為矩陣的線性運(yùn)算線性運(yùn)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 只有當(dāng)矩陣只有當(dāng)矩陣A 的的列數(shù)列數(shù)等于矩陣等于矩陣B 的的行數(shù)行數(shù)時(shí)，乘積時(shí)，乘積AB才有意義才有意義. 否則，不能相乘否則，不能相乘. 乘積矩陣乘積矩陣C 的行數(shù)與矩陣的行數(shù)與矩

5、陣A 的行數(shù)相同，的行數(shù)相同， 矩陣矩陣C 的列數(shù)與矩陣的列數(shù)與矩陣B 的列數(shù)相同的列數(shù)相同41010311132102201134AB 與與41010311132102201134AB 9219911 24241236AB 與與24241236AB 24243612BA 1632816 0000 矩陣的乘法矩陣的乘法不滿足交換律不滿足交換律. AB是是A左乘左乘B，BA是是A右乘右乘B. 如果兩個(gè)如果兩個(gè)n階方陣階方陣A,B，若，若AB=BA，稱，稱A與與B是是可可交換的交換的.010210,000000ABC , ,0000AB 0000AC ,ABACAO BC 矩陣的乘法矩陣的乘法不滿

6、足消去律不滿足消去律. 矩陣的乘法矩陣的乘法不滿足交換律不滿足交換律. 矩陣的乘法矩陣的乘法不滿足消去律不滿足消去律.22.5311.533112A 11yx032321.52.5a0cossin66sincos66R BRA 0.2330.6652.0980.3660.2993.5982.1162.3361.3662.482 a0圖形繞原點(diǎn)旋圖形繞原點(diǎn)旋轉(zhuǎn)了轉(zhuǎn)了30度度234121228A 1231231232344232282xxxxxxxxx 123xxxx 432b Axb kAAAA (AB)k = AkBk. (A+B)2 = A2+2AB+B2. Ak=O , A不一定為不一定為

7、O. 對(duì)角陣對(duì)角陣 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是對(duì)稱矩陣是對(duì)稱矩陣 AT = A. A是反對(duì)稱矩陣是反對(duì)稱矩陣 AT = A. A是方陣是方陣 (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 與與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 任何一個(gè)矩陣都可以經(jīng)過有限次任何一個(gè)矩陣都可以經(jīng)過有限次初等行變換初等行變換化為化為行最簡矩陣行最簡矩陣，且行最簡矩陣，且行最簡矩陣唯一唯一.123212024231373283023743rrr 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 任何一個(gè)矩陣都可以經(jīng)過有限次任何一個(gè)矩陣都可以經(jīng)過有限次初等行變換初等行變換化為化為行最

14、簡矩陣行最簡矩陣，再經(jīng)過有限次初等列變換化成，再經(jīng)過有限次初等列變換化成等價(jià)等價(jià)標(biāo)準(zhǔn)型標(biāo)準(zhǔn)型.011111( , )1101E i j 11( ( )11E i kk 11( , ( )11Eki j k A A 111212122212nnnnnnaaaaaaDaaa 1212nppnpaaa共共 n!個(gè)個(gè)( 1) a12a21a11a2211122122aaaa 111213212223313233aaaaaaaaa a11a22a33 a12a23a31 a13a21a32 a11a23a32 a12a21a33 a13a22a31 對(duì)角線對(duì)角線法則法則= a11 a22ann .112

15、12212000nnnnaaaaaa11121222000nnnnaaaaaa三角形三角形行列式行列式= a1n a2 n-1an1 .1122000000nnaaa12110000000nnnaaa 對(duì)角型對(duì)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 將將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 行最簡形行最簡形矩陣矩陣12210132 ,0121312AB 174298110 ,1153165A

20、X 12210132 ,0121312AB 122102132 ,113213AC 111213142122232431323334aaaaaaaaaaaa11133133aaaakkmnC C401204180322A 401204180322A 401204180322A 22001 規(guī)定規(guī)定零矩陣零矩陣的秩為的秩為0. 若矩陣若矩陣A中有一個(gè)中有一個(gè)s階的子式不為零，則階的子式不為零，則r(A) s . 若矩陣若矩陣A中所有中所有t 階的子式全為零，則階的子式全為零，則r(A) t . 若若A為為n階矩陣，則階矩陣，則A的的n階子式只有一個(gè)階子式只有一個(gè), 即即|A|當(dāng)當(dāng)|A|0 時(shí)，時(shí)，r(A)= n ,可逆矩陣（非奇異矩陣）又可逆矩陣（非奇異矩陣）