《多元统计分析》课程教学资源（书籍文献，矩阵论）王松桂等（2004）线性模型引论，科学出版社

82.2退说动多物- 21-阵疾对2.2.1rk(A) =合A为mxn所阵，A=PQ模为mxm，nxn的称别取P和Q阵函述I.BA-:DL别取B,C和D为适单数的任意所阵适模合X为A的解这有若分I0AXA=APQXPQXP000疾记B11B1QXP-B22B21示分面Ir2B11H1000B12I于是，AXA=B12，B21和B22任意B21B22三模论2.2.1(1)对任意所阵A，A-物是应在的第将模重(2) A-里阵直A-=A-1;A为称(3) rk(A-) ≥ rk(A) = rk(A-A) = rk(AA-):总家(4) R M(B),C M(A),M(C) C M(A),均CA-B与A-的全分计然由模的因解这，就三结论或但待因心2.2.1也厂可四要意若孝地计后论合“之M(B)C,M(A),M(C) CM(A)程所阵T1,T2B项地标阶独标模称AT1,C=A'T2,所要结论立

§2.2 ã✰ä✰å✰✷✰✹ · 21 · ✈✁✇ 2.2.1 ✯ A ➜ m × n ①➐➨ rk(A) = r. ✥ A = P   Ir 0 0 0   Q, ❍✁■ P í Q ✡✁❘➜ m × m, n × n ➥✁✔✁❾➐➨ ★ A − = Q −1   Ir B C D   P −1 , ❍✁■ B, C í D ➜❭➞✁➼◆❭Ð➥ø❭ù✁①➐❭➓ ⑧✁⑨ ✯ X ➜ A ➥❭ã❚ ❾❭➨ ★ó AXA = A ⇐⇒ P   Ir 0 0 0   QXP   Ir 0 0 0   Q = P   Ir 0 0 0   Q ⇐⇒   Ir 0 0 0   QXP   Ir 0 0 0   =   Ir 0 0 0   . ✥➮ QXP =   B11 B12 B21 B22   , ★✁❸➬ ⇐⇒   B11 0 0 0   =   Ir 0 0 0   ⇐⇒ B11 = Ir . ❰×❭➨ AXA = A ⇐⇒ X = Q−1   Ir B12 B21 B22   P −1 , ❿ ÷ B12, B21 í B22 ø❭ù➓ ✜✁➁➓ ➂✁➃ 2.2.1 (1) ê ø❭ù✁①➐ A, A− æ×✱ ý ➥✁❄ (2) A− ❜❭➚ ⇐⇒A ➜✁✔✁❾✁➲➐❭➓ ↕➽ A− = A−1 ; (3) rk(A−) ≥ rk(A) = rk(A−A) = rk(AA−); (4) ✥ M(B) ⊂ M(A),M(C) ⊂ M(A0 ), ★ C 0A−B þ A− ➥✁ç✁è✁❇✁❃➓ ⑧✚⑨ éÜ✚❪➈➹➈➤✲❐❰ ❙✚➀ 2.2.1 ➆ ã❚ ❾➈➥❙✚❚✚❁✉➓✣❑✚↔❪ ❷➧➡ ù ✉❭➨✣❡✯❪✁❫ M(B) ⊂ M(A),M(C) ⊂ M(A0 ) ê✁ë✁ì➨ ✱ ý✁①➐ T1, T2 ❀✁❁ B = AT1, C = A0T2, í ✔✜P✫ä➧❭➹❭➤➓ ✜✁➁➓

.22.量张成空间交投影变子系标售告2.2.2A,边退文缺将依(1) A(A'A)-A"线司（A'A)-研究反诸(2)A(AA)-AA=A,A'A(AA)-A=A"详系根B.换分A'=AAB写形2.1.1M(A) = M(A'A),（1与佳度上文品将待立但A(A'A)-A'=B'AA(AA)-A'AB=B'A"AB,(A"A)-反表诸直维有退缺第=A(AA)-看-差:不退(2)线司：FF=0.待立佳线但差子示有且有差萌茶表表非F = 0.分独缺维、他余身同间低同2.2.2历值但但值化著不表天缺系别皆边两宗退重表由缺家如佳桌赖线司很客真倾混全过混说不分IL文非品表特子皆逆白英2.2.2Ar=b变客真倾但范子退依(1) 标,线司A-，T=A-b”变混边缺缺机均(2)某一客真倾AL=0z变今但A-或混变工三(I- A-A)z,子蒙退边当时一依变令佳线司边便缺或混变(3) Ar =bT=A-b+(I-A-A)z,(2.2.3)子缺退线司但变，令A-变佳商表边、便边重情特详，当时金根边子写形(1)与但标zo,换Aro =b.A-, A(A-b) -七度边AA-Ao=Aro=b.资A-b变混表特缺子-0(2)To变Ar：混但资Aro=0,边原具ro = (I - A-A)ro + A-Aro = (I - A-A)ro,缺表示子顺子缺资混上(I-A-A)z表化性但标费A(I-A-A)z=(A-2边边逼AA-A)z=0,(I-A-A)z”变混表子家退详缺子家七佳(3)线司A-，与(1)1=A-b变客真倾Ar=b小混表重力缺数缺间某与(2)客真倾A=0(I -A-A)z变某或混表很客真倾2.元详的否否有缺或混表但1+2变A=b销表混身的佳品细重缺特皆逆白美2.2.3银但Ar =客真倾但表b≠0，hA-上添A说退如干原具的，缺对交混表客真倾线司主但=A-b向名早缺有第有偏子家写形交或倾向表商但值A-b变Ar=b标A-与非交集理子子缺机有根子家2混但叉就佳表商明但值Ar=b混0，X标品金锌边上缺子家退根AA-，换ro=A-b.与(2.2.3)但线司G密20，换分上T0 =Gb + (I - GA)z0

· 22 · ✴✰✵✰✶✸✷✰✹✰✺✰✻✰✼✰✽✰✾✰✿ ➂✁➃ 2.2.2 ê ø➚ ①➐ A , (1) A(A0A) −A0 þã❚ ❾ (A0A) − ➥✁ç✁è✁❇✁❃✁❄ (2) A(A0A) −A0A = A, A0A(A0A) −A0 = A0 . ⑧✆⑨ (1) ❉ ❙✆➀ 2.1.1 ➻ M(A0 ) = M(A0A), ❶ ✱ ý✆①➐ B, ❀✆❁ A0 = A0AB. ❰×❭➨ A(A0A) −A0 = B0A0A(A0A) −A0AB = B0A0AB, þ (A0A) − ❇✁❃➓ (2) ➮ F = A(A0A) −A0A − A, î ➭➈ã❚ ❾➈➥❙✚❚➨✣✔▼ ✛✚✜ : F 0F = 0. ❰× F = 0. ❑➚✁➬❁✜➓✣ïì ✔✜❑Ú✁➬➓ â❭➤ 2.2.2 ➥❭➹❭➤✁ðt➥➧❭➨ ▼ ➠✃❭❐➧✁ñ✁ò❭➭✁✉➓ ♣➡➥ ï ç ❙ó➀õô ☎óö❝ó×➝ ➭ ã❚ ❾ ①➐➸➺ ❂✘➯➲ó➲ó➳Ôó❝è➥ Ø Ù➓ ✈✁✇ 2.2.2 ✯ Ax = b ➜❭➚✁❂✘➲✁➳Ô ➨ ★ (1) ê ø➚❭ã❚ ❾ A−, x = A−b ÷ ➜❝ ❄ (2) ø✁ù➲✁➳Ô Ax = 0 ➥➘✁❝➜ x = (I − A−A)z, ❍✁■ z ➜ø❭ù➥ ÑÓÒ➨ A− ➜ø❭ùPú❦❙➥❭➚❭ç❭ã❚ ❾✁❄ (3) Ax = b ➥➘✁❝➜ x = A −b + (I − A −A)z, (2.2.3) ❿ ÷ A− ➜ø➚ ú❦❙➥❭ã❚ ❾❭➨ z ➜ø❭ù⑩ÑÓÒ➓ ⑧✚⑨ (1) ❉ ❂✘➲✚❡✯✚➻➨ ✱ ý x0, ❀ Ax0 = b. ❶ ê ø➚ A−, A(A−b) = AA−Ax0 = Ax0 = b. ö A−b ➜❝ ➓ (2) ✯ x0 ➜ Ax = 0 ➥ø➚ ❝ ➨➅ö Ax0 = 0, û✁ü x0 = (I − A −A)x0 + A −Ax0 = (I − A −A)x0, öø➚ ❝ ✭✁ý (I − A−A)z ➥✁➷✁➬➓ ñ➴✁þ➨➅êø➚❭➥ z, ➄ A(I − A−A)z = (A − AA−A)z = 0, ❶ (I − A−A)z ÷ ➜❝ ➓ (3) øý ❙➚❜ç❜ã❚ ❾ A−, ❉ (1) ➻ x1 = A−b ➜✆➲✆➳Ô Ax = b ➥❜➚❜ç◗✆❝➓ ❉ (2) ➻ x2 = (I − A−A)z ➜ø✁ù➲✁➳Ô Ax = 0 ➥➘✁❝➓ ➅✁ðø✁ù➯❭➲✁➲✁➳Ô ➥ ❝ ➹✁✖❙✁➀➻➨ x1 + x2 ➜ Ax = b ➥➘✁❝➓ ✜✁➁➓ ✈✁✇ 2.2.3 ✯ Ax = b ➜✁❂✘➯❭➲✁➲✁➳Ô ➨❦✂ b 6= 0, û✁ü➨❦➼ A− ý✁ÿ A ➥ ä❭óã❚ ❾➽➨ x = A−b ✖Õ ➝✁￾✁➲✁➳Ô ➥✁✍✁✂❝ ➓ ⑧✁⑨ ✜P✫ ❉ ï ✂ ✡❭Ô❭Õ➓ ❿➚❭➨➅➧✜ê✁❲➈➚➈ç A−, x = A−b ➜ Ax = b ➥ ❝ ➨✣❍☎✄ýé➚ ❙✁➀⑩÷✜P✫➴ ➝ ➓ ❿Ú❭➨➅➧✜ê Ax = b ➥ø➚ ❝ x0, ÷✱ ý➚❭ç A−, ❀ x0 = A−b. ❉ (2.2.3) ➻➨ ✱ ý A ➥❭➚❭ç❭ã❚ ❾ G ➆ z0, ❀✁❁ x0 = Gb + (I − GA)z0

82.2退说动多物-23 -系无物应用b±0. 测且称上U=20(66)-16.已严所U20=Ub.设项地TO= Gb+(I - GA)Ub= (G+(I - GA)U)6 Hb)行向-三根从H=-G+(I-GA)U.研A-.因心回H收地根现所历根因心2.2.2历（3）则因心2.2.3法所别研因心严Urquart已1969裱程系究历质恭客重都点五(2.2.3)"A-A)z收且A-严因历且（I非间理R从重因心2.2.3”严性房且严某囍且某历别点五们名应术上本粮非间否便记计方并角关历含同”何但值种但镇上来模根系臂摩奇量归系模何但含同所应历历一即较关非统历藝不龈来上版或粮计的则、心而，过并或称历因历一即函是若立地称阵记合对2.2.4A1lA12A=A21A22模根疾称[A11/+0,分A11A12Ai1+A-lA12A221A21A-l-A-A12AA-1A2.1A22A22.1A421Ai1A21(2.2.4)疾[A22|±0,分Ail.2-A112A12A22A-1(2.2.5)-A2A21A-12A2+A-A21A-12A12A2从，A22.1=A22-A21A-A12,A11.2=A11-A12AA21.疾[A11l + 0,解这八名分I0A11A12-AiA12A110I1-A214llA2100IA22A22.1(2.2.6)模根计示三标口模系。单向A22.1历称其且客所非它地-10AiiA127AilA12Ail010I0A21A22A22.1-A21Ail1AII +AI"A12A.1421AIl-A'A12A22.1A22.1A22.1A21Ail

§2.2 ã✰ä✰å✰✷✰✹ · 23 · ➄ b 6= 0, ❶æ✱ ý✁①➐ U, ❀✁❁ z0 = Ub. ✆❈➨✣✔✁ý U = z0(b 0 b) −1 b 0 . ❰× x0 = Gb + (I − GA)Ub = (G + (I − GA)U)b 4 = Hb, ❿ ÷ H = G + (I − GA)U. ✙✁✛✁✜ H ➜❭➚❭ç A−. ❙✁➀✁❁✜➓ ❍❭ç❙✁➀× ❉ Urquart ❰ 1969 ➓✁✝❮ ➥➓ ❙✁➀ 2.2.2 ➥ (3) í✁❙✁➀ 2.2.3 ❥❮ ➝✁❂✘➯❭➲✁➲✁➳Ô✁❝è❭➥ï❭ð➸❭➺❭➓ ý (2.2.3) ÷ ➨ A− × ú❦❙➥❭➨ (I − A−A)z ➜ ø❜ù✟✞➓ ❋ý✆❙✆➀ 2.2.3 ÷ ➨ A− ×✟✠❜➥❜➨❈×ø❜ù➥➓ ❍ ï❜ð➸❜➺✁✡❜ó❿➲✁☛ú✟☞➨ ý ▼ ➠❭➥❭➢❭➤÷✃❭❐➧s✁t➭✁✉❭Ö❐❭➓ ♣➡✃❜❐➢❜➤✡✟✌✆①➐➥❜ã❚ ❾➓ ➭✆➯➏✆➐✁❾①➐✆✱ý ➥✟✍✁✎❜➨ à❭➠➩✁ï✟✏➥ ✑✁✒í✁☞✁➀✁✓✁✔Ø✁✕❭➟❭➭✁✉✲✔✁❾❭➥✁✍✁✎➓ í✁❁✉✡✁✌ã❚ ❾❭➥❭➹❭➘➓ ✈✁✇ 2.2.4 ✯ A =   A11 A12 A21 A22   ✔✁❾➓✣✥ |A11| 6= 0, ★ A −1 =   A11 A12 A21 A22   −1 =   A −1 11 + A −1 11 A12A −1 22.1A21A −1 11 −A −1 11 A12A −1 22.1 −A −1 22.1A21A −1 11 A −1 22.1   . (2.2.4) ✥ |A22| 6= 0, ★ A −1 =   A −1 11.2 −A −1 11.2A12A −1 22 −A −1 22 A21A −1 11.2 A −1 22 + A −1 22 A21A −1 11.2A12A −1 22   , (2.2.5) ❿ ÷ A22.1 = A22 − A21A −1 11 A12, A11.2 = A11 − A12A −1 22 A21. ⑧✁⑨ ✥ |A11| 6= 0, ★ó   I 0 −A21A −1 11 I     A11 A12 A21 A22     I −A −1 11 A12 0 I   =   A11 0 0 A22.1   . (2.2.6) ↕ ➬✜P✫➝ A22.1 ➥✁✔✁❾❭➲➓ ï✁✖✁✗❾ ①➐➨ ✘✁✙❁ ✉   A11 A12 A21 A22   −1 =   I −A −1 11 A12 0 I     A −1 11 0 0 A −1 22.1     I 0 −A21A −1 11 I   =   A −1 11 + A −1 11 A12A −1 22.1A21A −1 11 −A −1 11 A12A −1 22.1 −A −1 22.1A21A −1 11 A −1 22.1  

掌握真.24.道章多物模重向单三标用记因心的后非统的法称用澳引对 测果A-1或应在，自然统的它的广结果，我们有测“若疾阵模应在，对2.2.5（所阵的广(1)Ai1分若函是A-1+A-A12A221A21Ai1Ai1A12-Ai"A12A22.1(2.2.7)A22-A22.1A21AilA22.1A21疾(2)A2 应在，分An.A12Ai1.2-Ai1.2A12A22(2.2.8)-A22A21A11.2A2+A2A21Ai1.2A12A22A21A22疾(3)A11A12A:≥0,A21A22分Aii + AiiA12A52.1A21Ail-AiiA12A22.1(2.2.9)A-A22.1A22.1A21Ail子Ai1.2-A11.2A12A22A-(2.2.10)-A22A21A12A2+A22A21A12A12A2从中 A22.1 = A22- A21AiiA12, A11.2= A11 -A12A22A21三标标解这我们(1)和(3),(2)的与(1)后了回三元示仍成(1).单Ai1应在直，(2.2.6)实：B=PCQ,P、Q于是模很推误例标，称S压.有B-=Q-10-P-分-A-1A12A1041At0-A21AilA220IA210A22.1A-lA12Au-A21Ail10A22.11

· 24 · ✴✰✵✰✶✸✷✰✹✰✺✰✻✰✼✰✽✰✾✰✿ ➭✁✘✁✍ï✁✏➥✁➲ì ✔▼ ✜P✫❙✁➀➥❭➠✁✙✁✂✡ ➓ ❈➘ A−1 ✲✱ ý ➨ Þáà✏✁✒Ö❭➥❭ã❚ ❾➓ ê ↕ ➨ ✃❭❐❭ó✁❈✁♣➹❭➘➓ ✈✁✇ 2.2.5 (✡✁✌✁①➐➥❭ã❚ ❾ ) (1) ✥ A −1 11 ✱ ý ➨ ★   A11 A12 A21 A22   − =   A −1 11 + A −1 11 A12A − 22.1A21A −1 11 −A −1 11 A12A − 22.1 −A − 22.1A21A −1 11 A − 22.1   . (2.2.7) (2) ✥ A −1 22 ✱ ý ➨ ★   A11 A12 A21 A22   − =   A − 11.2 −A − 11.2A12A −1 22 −A −1 22 A21A − 11.2 A −1 22 + A −1 22 A21A − 11.2A12A −1 22   . (2.2.8) (3) ✥ A =   A11 A12 A21 A22   ≥ 0, ★ A − =   A − 11 + A − 11A12A − 22.1A21A − 11 −A − 11A12A − 22.1 −A − 22.1A21A − 11 A − 22.1   (2.2.9) ➚ A − =   A − 11.2 −A − 11.2A12A − 22 −A − 22A21A − 11.2 A − 22 + A − 22A21A − 11.2A12A − 22   , (2.2.10) ❿ ÷ A22.1 = A22 − A21A − 11A12, A11.2 = A11 − A12A − 22A21. ⑧✁⑨ ✃❭❐❷✜P✫ (1) í (3),(2) ➥✜P✫þ (1) ✚✁✛➓ ➯✜ (1). ➼ A −1 11 ✱ ý➽➨ (2.2.6) ➬ ü❭Õ✁❢➓ ❰×✁➪✁✜✁❤Ï : B = P CQ, P Û Q ✔✁❾❭➨ ★ B− = Q−1C −P −1 (✜P✫✁✐❊✁♦Ù ), ó   A11 A12 A21 A22   − =   I −A −1 11 A12 0 I     A11 0 0 A22.1   −   I 0 −A21A −1 11 I   =   I −A −1 11 A12 0 I     A −1 11 0 0 A − 22.1     I 0 −A21A −1 11 I  

价，多章-25.$2.2AS他证如现但他但利不基知：A1l00A22.1系立准标角0An10A22.1缺退说三况由部家系义逆表开但资里表所面再一(3).角A≥0,于应B= (所(B1 :B2),里B(BiBB2AnlA12A=B'B-BB2A21A22B,B1缺低同2.2.2(2),名程(2.2.11)A21AiA11 = B,Bi(B,B)-B,B1 = B,Bi = A21(2.2.12)A11AiA12=B,B1(B,B1)-B,B2=B,B2=A12待立但。(2.2.6)质类似但名V1A110A110A12T-AiA1210A21A220A22.1-A21Aii(2.2.13)数求缺由就基知据不非但称里质元诸独:An1A12TAii0T0-AiiA120IA21A220A22.11-A21Ai1系部说三三模示表方但资里说达质表不类似非称可明表因析所年换他，说独降缺退缺子聚若切称用看所但他但说聚因析义递1部表用但因后缺退证打，水岩或影响(2.2.7)~(2.2.10)，映析系变右端立义逆表说达点他缺由缺缺他说等但但映不表用变，很模型估计析同，和A-植关易择反心时诸等金表缺然便便示子A≥0称用因析讨减弱翻变但M(A12)CA1A2应M(At)CM(Ai）称低所A售412=A12AaA=A21,待M(Au1)立但(2.2.13)向录表用但(2.2.9)（2.2.10）向表于量o类米

§2.2 ã✰ä✰å✰✷✰✹ · 25 · ❍✁■❭➨ ✃❭❐î ➭❭➝✁❤Ï✁❩   A −1 11 0 0 A − 22.1   ×✁✢❭ê✁✣➐   A11 0 0 A22.1   ➥❭ã❚ ❾➓✣➩❸➡❭Ü❭ç①➐ î✁✤✁þ➨➅ö❁ä✜➓ ✥✜ (3). ➄ A ≥ 0, ❶ ✱ ý✁①➐ B = (B1 . . . B2), ❀✁❁ A = B 0B =   B0 1B1 B0 1B2 B0 2B1 B0 2B2   =   A11 A12 A21 A22   , ❉ â❭➤ 2.2.2 ➥ (2), ó A21A − 11A11 = B 0 2B1(B 0 1B1) −B 0 1B1 = B 0 2B1 = A21, (2.2.11) A11A − 11A12 = B 0 1B1(B 0 1B1) −B 0 1B2 = B 0 1B2 = A12. (2.2.12) ❰×❭➨ í (2.2.6) ❂✚✁✛➨ ó   I 0 −A21A − 11 I     A11 A12 A21 A22     I −A − 11A12 0 I   =   A11 0 0 A22.1   . (2.2.13) ➅ ↕ ❤ Ï✁➆➭þé➡✁✘✁✍✁❂ï➥✁➲ì ➨✣✔❁   A11 A12 A21 A22   − =   I −A − 11A12 0 I     A − 11 0 0 A − 22.1     I 0 −A21A − 11 I   . ✦ ↕Ü ①➐❂î ➨➅ö❁ä✜➓ ➭✚✁✛➲ ì ✔✜❑Úð ➸✁✦➬➓ ❙✁➀✜✁➁➓ ❰ ❙✁➀✜P✫➴ ➳✁✔▼✁✧❮ ➨ ✃❭❐❭ä✗ ✉❭➥❭ã❚ ❾ ❷× A− ➥❭➚✁✂✡ ➓ ➄↕ ➨ ❙ ➀⑩÷➥ A− ➸✁✦➬ (2.2.7)∼(2.2.10), ➟ ➀✁❝➜❹✁★× A ➥❭ã❚ ❾➓ ❍❭➚✁✩❭➴✲✁✪✁✫ ✃❜❐➠❜➡❜➥❜➟❜➭➓ ➄➜ý ➯❭➲✟✬✁✭✟✮✁✯➀➤ ÷ ➨ ✃❭❐❜ä❃✁✰❜➥Ò ✭ þ A− ➥✆ç✆è✆❇ ❃➓ ❙✆➀➥✆❪✆❫ A −1 11 ➚ A −1 22 ✱ ý ➚ A ≥ 0 ➫ ✔▼➬❜➚✆❞✟✱✟✲➓ ➄➜❜➨ ❉ M(A12) ⊂ M(A11) í M(A0 21) ⊂ M(A0 11) ✔➈â❮ A11A − 11A12 = A12 í A21A − 11A11 = A21, ❰ ×❭➨ (2.2.13) Õ✁❢➓ ➄↕ ➨ (2.2.9) í (2.2.10) ￾❭Õ✁❢➓ ❶✁❁