关于矩阵秩不等式问题的证明与应用.pdf

Advances in Applied Mathematics A?,2024,13(4),1433-1447Published Online April 2024 in Hans.https:/www.hanspub.org/journal/aamhttps:/doi.org/10.12677/aam.2024.134134u?K?yA=?=vF2024c3?19FF2024c4?17FuF2024c4?24F?|?dIO/)n!?4|!?C!5m?!g|?:)X!?n?y?Ac?yAThe Proof and Application of InequalityProblem Related to Matrix RankXueping HeXigu District Lintao Street School,Lanzhou GansuReceived:Mar.19th,2024;accepted:Apr.17th,2024;published:Apr.24th,2024AbstractIn this paper,the proofs of some common matrix rank inequalities are studied main-ly using equivalent normal form decomposition or full rank decomposition theorem,:.u?K?yAJ.A?,2024,13(4):1433-1447.DOI:10.12677/aam.2024.134134maximal independent group of column vectors,block elementary transformation,thedimension of linear space and the basic solution system of homogeneous equations,and lists the application of some common matrix rank inequalities.KeywordsMatrix,Rank,Inequality,Proof,ApplicationCopyright c?2024 by author(s)and Hans Publishers Inc.This work is licensed under the Creative Commons Attribution International License(CC BY 4.0).http:/creativecommons.org/licenses/by/4.0/1.9(J1.1.?g?5?A?5?uS?3L-V?Cy3u?|=?n?A?2?=?2/A?+3n!E?X-?1,2?n?-53p?“!)A?+?2?AX?uEd!+n!?Jd!?!A+nE?Xv”-?3)A?2Au?9m!m!9m?X35n?55X3nU?*?U?d?5?|?55!|?4|!|m?55?d!g5|?)X!|?5L!)g5|?3,41.2.np?“;?:?nq?|?3kA-5?NX?5?Vg;?)?5kuS/)?Ky?0Bu?n?u?AO-?53)5|?5m:?X93)5A?m?X?kX2?A 5DOI:10.12677/aam.2024.1341341434A?-?Vgy“?A?-6?Vg?)u)5|=Ip43?K?z?!?f!?C?Vg6d?z?3p4?:?!?qC?Vg?-52A?+U?SB|?,kX?d 72.?95?k55 8?uf;_?,?,?u,?.5?9r A00B!=r(A)+r(B).5n 7?1?CUC?.5o 9r ABCB0!=r 0CB0!=r(B)+r(C).y ImA0In!0CB0!=ABCB0!,r ABCB0!r 0CB0!=r(B)+r(C).qr ABCB0!r(B)+r(C),Kr ABCB0!=r 0CB0!.n 2.1?A B n m?,K?r(A+B)r(A)+r(B).n 2.2?A m n?B n s?,K r(A)+r(B)n r(AB).n 2.3?A,B?,Kk?Xe.r(A)+r(B)=r A00B!r AC0B!.n 2.4A,B O s n n m?,AB=0.Kk?r(A)+r(B)n.DOI:10.12677/aam.2024.1341341435A?n 2.5?A B n m?,Kk r(A B)r(A)+r(B).n 2.6?A,B,C O mn,ns,st?,K r(ABC)r(AB)+r(BC)r(B).?n?e.,(2.2)?2.n 2.7r(A,B)r(A)+r(B).n 2.8?A n?,A?,Kk?r(A)=n,r(A)=n0,r(A)=n 11,r(A)n 1n 2.9?A,B n?K r(AB I)r(A I)+r(B I).n 2.10(?n?)A,D O n?m?.A _.B,C On m m n?,|?nk ABCD!A00D CA1B!,r ABCD!=r(A)+r?D CA1B?=n+r?D CA1B?.3.?y?yke8:!|?dIO/)n)ny?K?!|?C3y?K|?Cn!|?4|?u?1|?(1)?u?|?(?)?|?4|uy?y 10o!|g|?:)Xg Bx=0?:)X)nr(B)5?),(uy?!|5m?n 5m V?5C 3 V?1,2ne?B,K?u?B?,m?ug|BX=0?)m?,=dimIm=r(B),dimker=n r(B)DOI:10.12677/aam.2024.1341341436A?(,?K=z5m5C?K,?dimIm+dimker=dimV,dim(V1+V2)=dim(V1)+dim(V2)dim(V1 V2).Xx?,p?V1,V2O V?fm,1.?V U?5N?E,8!|?n 113?g?1(?)CA(m)?5?L3?pm?X1naCIm0PInABCD=AB0.3cmC+PAD+PB!p PA,PB k ABCD!,ABC+PAD+PB!?d?/ABCD!m2?Im00In!?AB+AQCD+CQ!Ny?CAmoN?5 ABC+PAD+PB!ABCD!AB+AQCD+CQ!.?AK?n?A _L1CK C Xe ABCD!AB0D CA1B!?I0CA1E!ABCD!=AB0D CA1B!UY AB0D CA1B!?C A K B=AB0D CA1BIA1B0I=A00D CA1B.DOI:10.12677/aam.2024.1341341437A?n?D _L D?B,C?ABCD!A BD1CB0D!A BD1C00D!,?,?ABCD?=|A|?D CA1B?=|D|?A BD1C?.4.(J?yn 2.1?y(|?C)A00B!A0AB!A0A+BB!r(A+B)r(A)+r(B).?C(?CUC?)1?11?K A?1?1?1?;1?1?1?1n?ABA0!0I0I!=0A+B0A!r(A+B)r 0A+B0A!r ABA0!=r 0BA0!=r(A)+r(B).qr(A)=r(A B)+B)r(A B)+r(B).n2.1 y.1?A B n m?,Kk r(A)r(B)r(A B)2?A B n m?,Kk r(A B)|r(A)r(B)|y r(A)=r(A B+B)r(B)+r(B+A)=r(B+A)+r(B),r(A)r(B)r(A+B).q r(B)=r(A A+B)r(A)+r(B+A)=r(B+A)+r(A),r(B)r(A)r(A+B),?r(A B)|r(A)r(B)|.ny|r(A)r(B)|r(A B).DOI:10.12677/aam.2024.1341341438A?n 2.2?y:ABA0In!IsOBIn!=0ABIn!,r(A)+r(B)r OA!r ABAn!=rAB0.=r(AB)+r(In)=r(AB)+n.r(A)+r(B)r OABIn!r ABA0In!=r AB00In!=r(AB)+r(In)=r(AB)+n.?n?C:?1K A?11?,?5p?5o.d,e AB=0,KkXe r(A)+r(B)n.?:|5m,?m=n=s?A/.?,n 5m V?5C3 12ne?O A,B.KdimIm+dimIm n dimIm()mindimIm,dimImky?V,()=(),=()Im,|()=(),K|Im Im(|)=Im()?5N?.d?dimIm=dimIm()+dimker().ker(),K()=0.=()ker,?ker()=ker Im.ddimker()dimker=n dimIm.2ym,d Im()Im,dimIm()dimIm.,ker,()=0 k()=0.=ker ker(),?n dimIm=dimker dimker()=n dimIm().l?dimIm()dimIm.?A,B n?,?.n 2.3?yeA,B k0,K(w,.d?r(A)=r 6=0,r(B)=r 6=0,K A,B Ok r,t?fMt6=0,Ms6=0,u A0CB!k r+t?f?Ms00Mt?6=0.DOI:10.12677/aam.2024.1341341439A?n 2.4?yAB=0,A0I0!IB00!=A0IB!,d2.3?r(A)+r(B)r A0InB!r A0In0!=r(In)=n.n 2.5?yr(A)+r(B)=r A00B!=r AA0B!=r AA B0B!r(A B).1?1n?O?C.?|?4|?A(12n),B(12n),KkA B=(1 1,2 2,n n);r(A)=s,r(B)=t i1is,j1jtO 1n,1n4|;1 1n ndi1in,j1jn5L r(A B)r(i1in,j1jn)s+t=r(A)+r(B).n|g|?:)Xe5|,|(A+B)x=0.|?AB!x=0,?du Ax=0 Bx=0.w,|?)|?),=|?)8u|?)8,?n r AB!n r(A B),l?r(A B)r(A)+r(B).n 2.6?ydu(5m)r(ABC)+r(B)=rABC00B=rABC0BCB=r0ABBCB r(AB)+r(BC),r(ABC)r(AB)+r(BC)r(B).?,?n?C;?,?1K A?1.DOI:10.12677/aam.2024.1341341440A?ABABCB0!IC0I!=AB0BBC!,r(BC)+r(AB)r BC0BAB!r BCABCB0!=r 0ABCB0!=r(ABC)+r(B).n 2.7?y(?|2.2,2.3 y)du(I,I)BOOA!=(B,A),5?r(A,B)r A00B!=r(A)+r(B).?|?dIO/)y?A,B?O r1,r2,K3_?P1,P2m?.Q1,Q2O t,s?,?A=P1 Ir1000!Q1,B=P2 Ir2000!Q2,PA1=Ir1000!,A2=Ir2000!.K(A,B)=(P1,P2)A100A2!Q100Q2!,d(2.2,2.3)r(A,B)rA100A2=r1+r2.n)y?A,B?O r1,r2,K3 m r1?P1,r1 t 1?Q1 m r2?P2,r2 s 1?Q2,?A=P1Q1,B=P2Q2,2d(A,B)=(P1,P2)Q100Q2!,9(2.2,2.3)?r(A,B)=r Q100Q2!=r1+r2.n 2.8?y?r(A)=n,du A=|A|A1_?.r(A)=n.?r(A)=n 1,kA?k n 1?f,A,o r(A)1.,?,?AA=|A|I=0,dn 2.7?r(A)+r(A)n,q r(A)=n1r(A)1,l?r(A)=1.?r(A)n 2,?A?k n 1?f,l?A=0,=r(A)=0.n 2.9?y A IB IA I0!0B0I!=0AB I0AB B!,DOI:10.12677/aam.2024.1341341441A?r(AB I)r 0B0I!=r 0AB I0AB B!A IB IA I0!=r0B IA I0=r(A I)+r(B I).n 2.10?y|?CUC?,?ABCD!L?C?A00D CA1B!,Kkr ABCD!=r(A)+r?D CA1B?.q?A_,kr(A)=n,kr ABCD!=r(A)+r(D CA1B)=n+r(D CA1B).5.?.?,?L?,$5B.K 1 12?A m n?B n s?,K r(A)+r(B)n=r(AB)?r AI0B!=r A00B!.yd IA0I!A0IB!IB0I!=0ABI0!.?r A0IB!=r 0ABI0!qr 0ABI0!=n+r(AB),r A00B!=r(A)+r(B).DOI:10.12677/aam.2024.1341341442A?K 2 13?A,B,C O m n,n l,l m?,?B?)B=P,=P?,Q 1?,K r(ABC)=r(AB)+r(BC)r(B)?3?X,Y?XAP+QCY=Ir.y?r(B)=r,B=PQ),kr(AB)=r(APQ)=r(AP)r(BC)=r(PQC)=r(QC).Kr(ABC)=r(AB)+r(BC)r(B)r(APQC)=r(AP)+r(QC)r.?y.K 3 14?A Pnn,f(x),g(x)P?.XJ f(x)g(x)p,Krf(A)+rg(A)=n+rf(A)g(A).yd(f,g)=1,3 u(x),v(x)Px,?u(x)f(x)+v(x)g(x)=1,dku(A)f(A)+g(A)v(A)=I,=?rf(A)+rg(A)=n+rf(A)g(A).K 4 15?A n?,K A3=A?7r(A)=r(A A2)+r(A+A2).y75:,d(I A)A(I+A)=0 9 FrobInius?.?0 r(I A)A+rA(I+A)r(A),=r(A)r?A A2?+r?A+A2?.,dr?A A2?+r?A+A2?r?A A2?+?A+A2?=r(2A)=r(A),=?75.5,e r(A)=r(A A2)+r(A+A2),?r(A)=r,A?)A=HL,K3 X,Y(2X)H=Ir,L(2Y)=Ir.KX(I A)H+L(I+A)Y=(XH+LY)(XHLH LHLY)=Ir 0=Ir.dK 2?r(I A)A(I+A)=r(I A)A+rA(I+A)r(A)=0,DOI:10.12677/aam.2024.1341341443A?=?(I A)A(I+A)=0,l?k A3=A.6.?K?AO|?C!g|:)X!4|!5m?!?ny?K?A 16.1 A,B n?.K r(A ABA)=r(A)+r(E BA)n.y A00I BA!AABAI!A ABA0BAI!A ABA00I!.y?IA0I!I0BI!A0AI BA!II0I!I0BAI!=A ABA00I!.2 A n?,yr(A)+r(A3)2r(A2).ydu r(A3)=r(AAA),qn 2.6?r(AAA)r(AA)+r(AA)r(A),?.3 A,B n?,AB=BA=0,r(A2)=r(A),y r(A+B)=r(A)+r(B).ydu r(A2)=r(A),Kkr?A2?=r(A)r(A)=r?A2?=r?A3?=Iky:A+B0A2+AB0!=A+B0A20!A+BA2+BAA2A3!A+BA2A2A3!BA20A3!5:1L1 1 1 A?1 2 1;1?L1 1?m A?1?;1nL1 2?mK P?1 1?.n?r(A2)=r(A2A),?|A2X=A k),?p,=k A2P=A.?r(A2,A)=r(A(A,I)r(A)=r(A2),r(A2,A)r(A2),?r(A2,A)=r(A2),AX2=A k)?.?1eCDOI:10.12677/aam.2024.1341341444A?A+BA2A2A3!BA0A3!.?Xr(A+B)=r BA0A3!r?A3?+r(B)=r(A)+r(B).?w,?.4?n|1s;1t;1s;1t?O r1,r2,r3,ymaxr1r2 r3 r1+r2.y ky()1s;?1s,1t5L,Kk r1 r3.n1t?1s,1t?5L,Kkr2 r3 maxr1,r2 r3.2y(m)k 1s,1t;?45|i1ir1,j1jr25L1s,1t;Kkr3 r1+r2.5?A,B P?n?AB=BA,y r(A)+r(B)r(AB)+r(A+B).y?|AX=0 BX=0?)mO V1,V2,ABX=BAX=0(A+B)X=0?)mO W1,W2,K V1 W1,V2 W2,l?V1 V2 W1,V1 V2 W2,|kdimV1+dimV2=dim(V1+V1)+dim(V1 V2)dimW1+dimW2.=(n r(A)+(n r(B)(n r(AB)+(n r(A+B).=r(A)+r(B)r(AB)+r(A+B).y?A00B!A0AB!AAAA+B!,(AB=BA,?AAAA+BA+B0AI=ABA0A+B.uDOI:10.12677/aam.2024.1341341445A?r(A)+r(B)=r A00B!r ABA0A+B!r(AB)+r(A+B).6 A s n?,y r?In ATA?r?Is AAT?=n s.y(|?n)InATAIs!|?n?In ATA00Is!InATAIs!I00I AAT!r I ATA00I!=r I00I AAT!.=s+r?I ATA?=n+r?I AAT?,r?I ATA?r?I AAT?=n s.7.o(?K?yA23?9?y|?dIO/)n!?4|!?C!5m?!g|?:)X!?n?8?1?yk?(?9?K/A?SKu?KyA?1?o(9?Ekv?:3?SUYS?)z1 S?.?59?yJ.p?,2020,23(4):96-99.2 M?.?9AJ.I“?:g,2010,12(5):19-21.3 u.1?F1/?35“?A93MATLAB?yJ.M?,g,2015,31(3):90-94.4?.|?#)o?J.?w“;?,g,2013,15(1):1-3,73.5?f.p?“SK8M.LH:E?,2002.6 az.;#?M.LH:?,2017.DOI:10.12677/aam.2024.1341341446A?7 4(.p?“oES?KM.?:?,2018.8 a.p?“M.?:?p?k?,2003.9 I,D=.|?y?J.?,1999(4):61-62.10?.5“21-Vp?A.5y?M.?:z?,2011.11 A7?,xg.?59?()?yJ.?,2023,42(3):27-33.12?Gp.ua?5PJ.E?,2004,27(3):322-323.13 MI?.?Frobeniusn?5PAJ.?:g,2005,19(2):3-5.14?.p?“#M.LH:?k?,1989.15?.ueZ?J.?“?:g,2010,30(1):1-4.16 o),?I.5“M.:IE?,1989.DOI:10.12677/aam.2024.1341341447A?