《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

-26.量张成空间，交投影变系标售告2.2.3文AA21行M(A12)CM(A1u1),M(A21)CM(Ai1),范(2.2.9)(2.2.10）向况表A+的2.2.2较缺详家家子期退机表俱仅名反存快反存快A-但名已甲散同司A-线茶团福L退缺子家子效条饮罐值但能罚立往提线司比A-Moore-Penrose他缺字台圖竟2F沿无佳线但然关同能表但白行家误在特定义表子系A变2.2.2但X观诉平较文边(2.2.14)AXA= A, XAX = X, (AX)-AX, (XA) -XA缺退某最系最X变A变A+.名主范线司，(2.2.14)变PenroseMoore-Penrose文客真表系某属缺引根特英2.2.1（映变r,Amxn变rk(A)=，范或混）文上非规律家系Pmxm希Qnxn，换无为客0AQ',A=(2.2.15)P00缺统变AA商A,=diag(1,间量小凯复表,入),入> 0,i=1,2, ..r.>2金系最系根但写形费变AA变标无为客Qnxnt卜超度A20Q'A'AQ=00某示资变B=AQ.甲A20B'B =00机期第缺数皆家数家回归B无为但表就r或二变入1,...,入，关n-当时过系当时数根+变量表得立但无为客Pmxm,换分上当时当时1B=00

· 26 · ✴✰✵✰✶✸✷✰✹✰✺✰✻✰✼✰✽✰✾✰✿ ➂✁➃ 2.2.3 ê ①➐ A =   A11 A12 A21 A22   , ✥ M(A12) ⊂ M(A11), M(A 0 21) ⊂ M(A 0 11), ★ (2.2.9) í (2.2.10) Õ✁❢➓ 2.2.2 ß✁à✁á A + ❰ ❸Ú❭➥❭➢❭➤➻➨✝➚✁✳✁✴þ ã❚ ❾ A− ó❇✁➶➟ç➓ ý❍✁❇✁➶➟ç A− ÷ ➨ ó ➚❭ç A− ✵ ó◗✁✶➥✁ö❏ ➨✝Öí ×❭➩❭➫❭➚✤✁✷✁✝✉❭➥ Moore-Penrose ã❚ ❾➓ ❻ý ✃❭❐❥❮ ❱➬❭➥❙✁❚➨➅à❭➠❭➢❭➤❭Ö❭➥❭➚❭➪❭➲✁✌➓ ✈à 2.2.2 ✯ A ➜ø➚ ①➐➨ ✥ X ☎✁✝♣➾↔ç✁❪✁❫❩ AXA = A, XAX = X, (AX) 0 = AX, (XA) 0 = XA, (2.2.14) ★✢ ①➐ X ➜ A ➥ Moore-Penrose ã❚ ❾ , ➮➈➜ A +. ó➽✢ (2.2.14) ➜ Penrose ➲✁➳➓ ✸✁✇ 2.2.1(✹✁✺✁✻✁✡✁❝) ✯ ①➐ Am×n ➥❛➜ r, ➮❭➜ rk(A) = r, ★✱ ý❭ï ç ❱✁✟➲➐ Pm×m Û Qn×n, ❀ A = P   Λr 0 0 0   Q 0 , (2.2.15) ❿ ÷ Λr = diag(λ1, · · · , λr), λi > 0,i = 1, 2, · · · , r. λ 2 1 , · · · , λ 2 r ➜ A0A ➥✁ð✁✳◗ ➮➪➓ ⑧✁⑨ ➄➜ A0A ➜❭ê✁✢➐➨ ❶ ✱ ý✁❱✁✟➲➐ Qn×n, ✼ Q 0A 0AQ =   Λ 2 r 0 0 0   . ➮ B = AQ, ❸ ➬❭ö❭➜ B 0B =   Λ 2 r 0 0 0   . ❍✁✴ ✫ B ➥✁✩ ÑÓÒ✁✽❂❱✁✟➨✣✂é r ç✁✩ ÑÓÒ❳✁❨✡✁❘➜ λ1, · · · , λr, ➠ n − r ç ✩ ÑÓÒ➜✁✳ÑÓÒ➓➅❰×❭➨ ✱ ý➚ ❱✁✟➲➐ Pm×m, ❀✁❁ B = P   Λr 0 0 0  

价，多章-27-$2.2BA疾三模B=AQ，业里(2.2.15)类程归入，,入为A的物模高两证T#所A+“个引析，称用它放性利用产T会者对2.2.6(1)设A有,(2.2.15),分们CA+:p(2.2.16)C0(将(2)对任身所阵A.A+人单向现写这直，回(2.2.16)的右解(2.2.14).(1)然(2.2.14)的况个%(2)设X和Y和是A+,之简程X =XAX -X(AX)-XX'A-XX'(AYA)=X(AX)'(AY)'= (XAX)AY=XAY- (XA)'YAY - A'X'A'Y'Y-A'Y'Y-(YA)Y-YAY-Y.将证一性则一求了握真.而用为A+是一个章的A-，用月，，它了具有A-的一且性是往有而误描性是型论2.2.4(1) (A+)+ = A;(2) (A+) = (A")+;(3) I ≥ A+A;(4) rk(A+) = rk(A);(5) A+ =(A'A)+A' = A(AA')+:(6) (A'A)+ = A+(A')+;(7)设a为一非量向量,分α+= d/l;积(8)A为对归非阵，它称表为A:证现P为阵，A, = diag(>1,..., Ar),r =rk(A),成线分4-10prA+= P00证通过节与法些基实的一求和基于(2.2.16)

§2.2 ã✰ä✰å✰✷✰✹ · 27 · ✥ ❉ B = AQ, ❢✁❁ (2.2.15). ✜✁➁➓ ➘✁t✢ λ1, · · · , λr ➜ A ➥✹✁✺✁✻➓ î ➭✁❍❭ç❭➷➀ ➨✣✔▼ ✖✁â❭➲✁ö❥❮ A+. ✈✁✇ 2.2.6 (1) ✯ A ó✡✁❝➬ (2.2.15), ★ A + = Q   Λ −1 r 0 0 0   P 0 . (2.2.16) (2) ê ø➹ ①➐ A,A+ ❜❭➚➓ ⑧✁⑨ (1) ➱✘✁✙Ø✁✕✛✁✜➨ (2.2.16) ➥❹✁★☎✁✝ (2.2.14). (2) ✯ X í Y ✭❭× A +, ❉ (2.2.14) ➥↔ç✁❪✁❫➻ X = XAX = X(AX) 0 = XX0A 0 = XX0 (AY A) 0 = X(AX) 0 (AY ) 0 = (XAX)AY = XAY = (XA) 0Y AY = A 0X0A 0Y 0Y = A 0Y 0Y = (Y A) 0Y = Y AY = Y. ❍ í✜P✫➝✁❜❭➚❭➲➓ ➄➜ A+ ×❭➚❭ç◗✁✶➥ A−, ➄↕ ➨➅Ö✁✾❭➝ò❭ó A− ➥✁✍✁✂➻➲✁✌✁✿❭➨ ➫ ó✁♣✩ ➲✁✌➓ ➂✁➃ 2.2.4 (1) (A+) + = A; (2) (A+) 0 = (A0 ) +; (3) I ≥ A+A; (4) rk(A+) = rk(A); (5) A+ = (A0A) +A0 = A0 (AA0 ) +; (6) (A0A) + = A+(A0 ) +; (7) ✯ a ➜❭➚✁ð✁✳ÑÓÒ➨ ★ a + = a 0/kak 2 ; (8) ✥ A ➜❭ê✁✢✁➲➐➨➅Ö✁✔➸➜ A = P   Λr 0 0 0   P 0 , ❍✁■ P ➜❱✁✟➐➨ Λr = diag(λ1, · · · , λr), r = rk(A), ★ A + = P   Λ −1 r 0 0 0   P 0 . ❍❭➪✁❤Ï ➥✜P✫✭❭➶❰ (2.2.16), ❀➫ ✐❦❥✁❧✁♠➓

.28.量张成空间，交投影变行重详皆逆表由但标2.2.3客真倾Ar=b,o=A+b”变混表已佳2.2.2很缺缺量记品如机家表佳混品将于缺重回归不用皆逆英2.2.7但TO=A+b变很客真倾Ar=b混全，均表上金缺且写形或混说变与(2.2.3),Ar=bT=A+b+(I-A+A)z待立r2 =(A+b+(I-A+A)z)(A+b+(I-A+A)z)=ro?+ z(I-A+A)?z+2b'(A+)(I-A+A)z=Jlrol?+z(I-A+A)?z≥roll2(2.2.17)缺费z(I-A+A)?z≥0标匠(A+)'(I - A+A) = (A+)- (A+)A+A = 0立销装增有向况表但记= A+b.(2.2.17)向况（I-A+A)z=0T再他说中金缺退定义误缺差子误对但节：同地AX家颜在甲线司A-(2.2:14)家聚映退国家家街定且维子家家柜边悲茶字滑然但高势管况表佳归系诉达重金边退边机台退线司平仔映不向鼻表磊待很反但文则向基意或增如金除用且上节保.同要[16],表$2.3性之低英重父亲庭缺儿阵缺将对交记客，x2或费变名任很平仔物子正但费积商父立缺公机子容亲顺他般解品+或这但名待但但往记客映不表能佳同～记客Y金量记增缺子合历置表亲亲白最父定义行客A22=A范2.3.1A变记(idempotent matrix)A.nxn白较父统行亲缺复给矩变0英2.3.1记小1.缺有第机家,逆满但已论表知任缺亲白厦A,英2.3.2标文边父亲顺I-AA-(1) A-A,AA-A-A.立记表小二但A+AAA+I-A+A父立亲依顺I-AA+立记立亲最交行(2)A变标记*但范A+=A.逆满具有写形已佳线(1)，寻不佳2.3.1低同2.2.4(8).况分(2)2立究白行交英2.3.3Anxn(1)记但范tr(A)=rk(A)记 rk(A) +rk(I -A) = n(2) Anxn

· 28 · ✴✰✵✰✶✸✷✰✹✰✺✰✻✰✼✰✽✰✾✰✿ ❰ ❙✁➀ 2.2.2 ➚ 2.2.3 ➻➨➅ê✁❂✘➯❭➲✁➲✁➳Ô Ax = b, x0 = A +b ÷ ➜❝ ➓✣♣➡ ➥❙✁➀✁❁✁❂➝✁❍❭ç❝ ➥❭➲✁✌➓ ✈✆✇ 2.2.7 ý ❂✘➯❜➲✆➲✆➳Ô Ax = b ➥❝ è ÷ ➨ x0 = A+b ➜✆❳✆❨❴✟❃♠➓ ⑧✁⑨ ❉ (2.2.3),Ax = b ➥➘✁❝✔➸➜ x = A +b + (I − A +A)z. ❰× kxk 2 = (A +b + (I − A +A)z) 0 (A +b + (I − A +A)z) = kx0k 2 + z 0 (I − A +A) 2 z + 2b 0 (A +) 0 (I − A +A)z = kx0k 2 + z 0 (I − A +A) 2 z ≥ kx0k 2 . (2.2.17) ➄↕ (A+) 0 (I − A+A) = (A+) 0 − (A+) 0A+A = 0 í z 0 (I − A+A) 2 z ≥ 0 ê ø❭ù➥ z Õ✁❢➓ ý (2.2.17) ÷ ➨✣☞❭➱Õ✁❢ ⇐⇒ (I − A+A)z = 0 ⇐⇒ x = A+b. ✜✁➁➓ ❸➡✃➈❐➈ä➢➈➤➈➥➈ã❚ ❾ A− í A+, ×✚☎✚✝ (2.2.14) ❑➚✚❪í✍❄✂↔❪➈➥ï ç✟❅★ ✍✟✎➓ Þ✡à✃❜❐➫ ✔▼ ❙✆❚☎✁✝↔ç✆❪✁❫÷Óø➚❭ç❜Û ø❜ïç✆➚øÜ❜ç❭➥❜ã❚ ❾➓ ❉ ❰❍❜➪❜ã❚ ❾ ý ➯❜➲✟✬✁✭❜➥✁➏✆➐ ÷➟❜➭✲✁❆✆✡ã✁❇❜➨ ↕✟☞✁í✆✲✥✟❈❜➬❭➚✆❞❭➥ ➢❭➤❭➝ ➓ ❧✁♠✔✁Ý✁Þ✁➣✁Ô [16]. §2.3 ❉❋❊❍● ➍ ➄➜✁■✁☞✁➲✁❏í χ 2 ✡✁❑ó➱✁▲✁✓❭➥✁❃✁➵❭➨ ➄❋ý ➯❭➲✁✬✁✭æ✁▼❭Ð✁➀✁◆✯❭➥❿ Ö❭➚❭➪✡✁❖⑩÷➨P■✁☞✁➲✁❏✁✭ó➚ ❙ ➥❭➟❭➭➓P◗❭❰↕ ➨ ✃❭❐ý❍❭➚❭➫✟❘☎❙Ó➢❭➤✟■✁☞✁➲ ❏❭➥❭➚❭➪➥➧❭➲✁✌➓ ✈à 2.3.1 ✥➲✟❏ An×n ☎✆✝ A2 = A, ★✢ A ➜✟■✆☞✟❏ (idempotent matrix). ✈✁✇ 2.3.1 ■✁☞✁❏❭➥◗ ➮➪ ❷✕❭➜ 0 ➚ 1. ❍❭ç✁❤Ï ➥✜P✫➱✘✁✙➨ ❰✁❚❭➓ ✈✁✇ 2.3.2 ê ø❭ù➥①❏ A, (1) A−A, AA−, I −A−A, í I −AA− ✭❜×✟■✆☞✟❏➓ ◗✆❘➨ A+A, AA+, I −A+A, í I − AA+ ✭❭×✁■✁☞✁❏✁❄ (2) ✥ A ➜❭ê✁✢✁■✁☞✁❏❭➨ ★ A+ = A. ⑧✁⑨ ❰ ❙✁❚✘✁✙✁✛✁✜ (1), î ➭❙✁➀ 2.3.1 íâ❭➤ 2.2.4 ú (8), ❢✁❁ (2). ✈✁✇ 2.3.3 (1) ✥ An×n ■✁☞❭➨ ★ tr(A) = rk(A). (2) An×n ■✁☞ ⇐⇒ rk(A) + rk(I − A) = n

82.3性无回物- 29-模重值解这(1) 合 rk(A) = r,应在称P,Q,项分1Q.A:从Q1从值中方P,Q-记：P=(P:P)中P为nxr的所Q1为O函Q2重计值42T×n的所，于是A=PiQ1.归一面地程10100C0三于QiP=Ir:所用tr(A)=tr(PQ1)=tr(QP)=tr(I)=r=rk(A).(1)搏平三均4的”型性举1型,实(2)要性是式然的.I-A若面程自虽的性是,n = tr(In) = tr(In -A + A) = tr(In - A) +tr(A) = rk(In - A) +rk(A)总某台化若诺：合k(A)总岔巢Af=0有n-r个线性均的系，它们是对应于诸翠捕总量的只一T个线性心均的向量.线性rk(I-A)Ar:=工有r=n集描某台某总程总均的系，它们是对应于1的个线性均的向量：用为别n描描描向量线性均，于是A质具于模值即应在称P.项A=P00三模于A?=A阵值均对2.3.4合Pnxn为对归, rk(P) = r,八应在问为的Anx项型分P=A(AA)-1A值均值用P为对归型于应在解这R= (R1:R2),项地成线RiI0I.P=RR=(R)R2)RR=R(RR)-1RR20000

§2.3 ❯❲❱❲❳⑤✹ · 29 · ⑧✁⑨ (1) ✯ rk(A) = r, ★✱ ý✔✁❾✁➲✁❏ P, Q, ❀ A = P   Ir 0 0 0   Q. ✦ P, Q ✡✁✌✁❩ P = (P1 . . . P2), ❿ ÷ P1 ➜ n × r ➥①❏❭➨ Q =   Q1 Q2   , ❿ ÷ Q1 ➜ r × n ➥①❏❭➨ ❰× A = P1Q1. ❨➚✁➲❭➡❭➨ ❉ A2 = A, ❁ ✉   Ir 0 0 0   QP   Ir 0 0 0   =   Ir 0 0 0   , ❶ Q1P1 = Ir. ä✁▼ tr(A) = tr(P1Q1) = tr(Q1P1) = tr(Ir) = r = rk(A).(1) ❁✜➓ (2) ÷➧❭➲❭×⑩à❭➥➓ ❤ Ï✁❸➨ ❉ A ➥✁■✁☞❭➲➻➨ I − A ￾ ■✁☞➓ î ➭✁❩✜ ➴ ➥❭➲✁✌❭➨ ó n = tr(In) = tr(In − A + A) = tr(In − A) + tr(A) = rk(In − A) + rk(A). ñ➴✁þ➨ ✯ rk(A) = r, ★ Ax = 0 ó n − r ç❭➯❭➲✁❇✁❃❭➥❝ ➨➅Ö❐×❭ê❭➟❰◗ ➮➪ ✳❭➥ n − r ç❭➯❭➲✁❇✁❃❭➥◗ ➮ ÑÓÒ➓ ❉ rk(I − A) = n − r ➻➨ Ax = x ó r ç❭➯❭➲ ❇✁❃❭➥❝ ➨➅Ö❐×❭ê❭➟❰◗ ➮➪ 1 ➥ r ç❭➯❭➲✁❇✁❃❭➥◗ ➮ ÑÓÒ➓ ➄➜✁❍ n ç ◗ ➮ ÑÓÒ➯❭➲✁❇✁❃❭➨ ❰× A ❂✛ ❰   Ir 0 0 0   , ö✱ ý✔✁❾✁❏ P, ❀ A = P   Ir 0 0 0   P −1 . ❶ A2 = A. ✜✁➁➓ ✈✚✇ 2.3.4 ✯ Pn×n ➜➈ê✚✢❄■✚☞❄❏ , rk(P) = r, ★✱ ý❛➜ r ➥ An×r, ❀ P = A(A0A) −1A0 . ⑧✁⑨ ➄ P ➜❭ê✁✢✁■✁☞✁❏❭➨ ❶ ✱ ý✁❱✁✟❏ R = (R1 . . . R2), ❀✁❁ P = R   Ir 0 0 0   R 0 = (R1 R2)   Ir 0 0 0     R0 1 R0 2   = R1R 0 1 = R1(R 0 1R1) −1R 0 1

-30.量张成空间，交投影变积有机均余如RiRi= Ir.分表不最A=R1.佳他中特缺子家很重秩都鼎表比但同无为无为ERn,S变R很上乘要立广要常表标工或混yes, zest,(2.3.1)T=y+z.缺最行子阵y变r表P变n态客都但换分标范S甲无为TERm,(2.3.1)佳缺”歪支缺上广最S线Pa,范无为都表详米=危缺常顺直垂称茶子当但但标R都读都Anxm，换分S=M(A)如边别文，表由缺说维2沿但都完为#表说”品特一要白缺英2.3.5A变nxm都但PA变M(A)无为都但范PA=文当广要A(A'A)-A'某子子写形B 变都但换分M(B)=M(A)+，范标TERn，名或混文贺编上边机均缺数数α,βB变表=Aa+BB佳线但PAT=PAAα+PABβ=Aa般尚时顺子阵标向况表都PA都客真倾α.B无为文度广要PAA=A.(2.3.2)PAB = 0誉明客真低分但M(PA)CM(B)广M。待立但根都U.PA=AU.小上文暴子无皆逆缺客真但分U'A'A=A.2.2.3.U=(AA)-A.待立客真倾立但与佳增品PA=U'A"=A((A"A)-)A'A(A'A)-A"如子家机均有销卖(1）密_（(A'A)-)比变低同2.2.2(AA)-.佳，映不说维最立慧季究将退费变PA=A(A"A产A)但无为线司研究反表诸广要最白英2.3.6P变无为都←P变标现记都表特写形P变都但与甲佳但P=AAA)-A=M(A)无为最重要品浙表寻不低同2.2.2(2),名A(A`A)+A.标分究P? = A(A'A)-A'A(A'A)-A' = A(A'A)-A' = P.重有式重有5值资佳，销表表或2.3.4.分品a白英2.3.7都一标n态客都P变无为rERn,边广要l-Pr=infllz-ull,uEM(P)(2.3.3)重有，”值严表边上uEM(P),vEM(P)+，某,写形*y=u+u范u=Py-并边l a-u2=ll -Py ll2=lla-Pr+Pr-Py2

· 30 · ✴✰✵✰✶✸✷✰✹✰✺✰✻✰✼✰✽✰✾✰✿ ❍✁■❭➭✁✉❭➝ R0 1R1 = Ir. ✥✁❬ A = R1, ❙✁➀✁❁✜➓ ❻ý ✃➈❐➢➈➤❱✚✟❪❭❄✪➈í❭❱✚✟❫❭❄✪❏➓✣✯ x ∈ Rn,S ➜ Rn ➥➈➚➈ç➈➯➈➲✚✤➈➳ ➵➓ ê x ❊✡✁❝ x = y + z, y ∈ S, z ∈ S ⊥, (2.3.1) ★✢ y ➜ x ý S ❸ ➥❱✁✟✁❭✁✪➓✣✥ P ➜ n ◆ ➲✁❏❭➨ ❀✁❁ê❭➚✁✓ x ∈ Rn,(2.3.1) ❙ ❚ ➥ y ☎✁✝ y = Px, ★✢ P ➜ Ñ S ➥❱✁✟✁❭✁✪❏➓ ✃➈❐✚➻❄❴➨➅ê Rn ➥ø➚✚✤➈➳➈➵ S, ✭✚✔▼❄❵✉①❏ An×m, ❀✚❁ S = M(A). ä✁▼ ➨ ♣➡❭➥❙✁➀❥❮ ➝❱✁✟✁❭✁✪❏❭➥➸❭➺❭➓ ✈✁✇ 2.3.5 ✯ A ➜ n × m ①❏❭➨ PA ➜ Ñ M(A) ➥❱✁✟❄❭✁✪❏❭➨ ★ PA = A(A0A) −A0 . ⑧❭⑨ ➮ B ➜➻➚①❏➻➨ ❀❭❁ M(B) = M(A) ⊥ , ★ê ø➚ x ∈ Rn, ó✡❭❝ x = Aα+Bβ, ❍✆■ α, β ➜❜➞✆➼❜ÝÐ ➥✆✩ Ñ✗Ò➓ ➅ ❙✆❚➨ PAx = PAAα+PABβ = Aα, ê❭➚✁✓ α, β ✭ Õ✁❢➓ ❶✁❱✁✟✁❭✁✪❏ PA ☎✁✝①❏✁➲✁➳Ô    PAA = A, PAB = 0. (2.3.2) ❉ ❑Ú✁➲✁➳❭â❁ ➨ M(P 0 A) ⊂ M(B) ⊥ = M(A). ❰×❭➨ ✱ ý✁①❏ U, P 0 A = AU. ÑÜ ❑➚✁➲✁➳❭➨ ❁ U 0A 0A = A. ↕➲✁➳Ô ×✁❂✘➥❭➨ ❉ ❙✁➀ 2.2.3 ,U = (A 0A) −A 0 . ❰× PA = U 0A 0 = A((A 0A) −) 0A 0 = A(A 0A) −A 0 . ❍✁■❭➟❭➭❭➝❭â❭➤ 2.2.2 ú (1) ➆ ((A0A) −) 0 ü ➜❭➚❭ç (A0A) −. ❙✁➀✜✁➁➓ ➄➜ PA = A(A 0A) −A 0 þã❚ ❾✁ç✁è✁❇✁❃❭➨ ä✁▼ ❱✁✟✁❭✁✪❏❭×✁❜❭➚❭➥➓ ✈✁✇ 2.3.6 P ➜❱✁✟✁❭✁✪❏ ⇐⇒ P ➜❭ê✁✢✁■✁☞✁❏➓ ⑧❭⑨ ✯ P ➜ Ñ M(A) ➥❱❭✟❫❭❪✪❏➻➨ ❉ ❸➚ ❙❭➀➨ P = A(A0A) −A0 = A(A0A) +A0 , ê✁✢❭➲❁✜➓ î ➭❭â❭➤ 2.2.2 ú (2), ó P 2 = A(A 0A) −A 0A(A 0A) −A 0 = A(A 0A) −A 0 = P. ÷➧❭➲❁✜➓❜❛✡ ➲❭ö❙✁➀ 2.3.4. ✜✁➁➓ ✈✁✇ 2.3.7 n ◆ ➲✁❏ P ➜❱✁✟✁❭✁✪❏ ⇐⇒ ê ø❥ x ∈ Rn, k x − Px k= inf k x − u k, u ∈ M(P). (2.3.3) ⑧✁⑨ ➯✜ ÷➧❭➲➓ øý u ∈ M(P), v ∈ M(P) ⊥ , ➮ y = u + v, ★ u = Py. k x − u k 2 = k x − Py k 2=k x − Px + Px − Py k 2