(+)분류 : 가져온 문서/오메가
Inverse matrix, Inverse
어떤 행렬 곱에 대한 역원으로, 어떤 행렬 A의 역행렬이 존재한다면 그 행렬을 가역(Invertible) 또는 정칙(Nonsingular)이라 하고, 역행렬은 A^{-1}로 나타낸다. 가역행렬과 그 역행렬의 곱은 항상 단위행렬이다.
AA^{-1} = A^{-1}A=I
1. 역행렬을 가질 조건 ✎ ⊖
n차 정사각행렬 A에 대해, 다음 명제는 모두 동치이다. 단, F는 체이다.
- A의 역행렬이 존재한다.
- \\operatorname{rank} A=n 이다.
- A의 행은 선형독립이다.
- A의 열은 선형독립이다.
- \\det A \\ne 0 이다.
- A의 치역의 차원이 0이다.
- A의 영공간의 차원이 0이다.
- 임의의 b\\in F^n에 대해 Ax=b의 해가 존재한다.
- Ax=b의 해가 존재하면, 그 해는 유일하다.
- 임의의 b\\in F^n에 대해 Ax=b의 해가 유일하게 존재한다.
- Ax=0의 해가 x=0뿐이다.
- 0은 A의 고유값이 아니다.
2. 공식 ✎ ⊖
역행렬의 공식은 모두 다음과 같다.
여기서 \\operatorname{adj}A는 A의 수반행렬, \\det A는 A의 행렬식을 말한다.
A^{-1} = \\frac{1}{\\det A} \\operatorname{adj}A
여기서 \\operatorname{adj}A는 A의 수반행렬, \\det A는 A의 행렬식을 말한다.
2.1. 이차 정사각행렬 ✎ ⊖
비교적 쉬운 이차 정사각행렬의 역행렬이다.
- \\displaystyle A_{2\\times 2}^{-1}=\\begin{bmatrix}a & b\\\\ c & d\\end{bmatrix}^{-1} = \\frac{1}{\\det A}\\operatorname{ adj} \\displaystyle A=\\frac{1}{ad-bc} \\begin{bmatrix}d & -b\\\\ -c & a\\end{bmatrix}
2.2. 삼차 정사각행렬 ✎ ⊖
- \\displaystyle A_{3\\times 3}^{-1}=\\begin{bmatrix} a & b & c \\\\ d & e & f \\\\ g& h & i\\end{bmatrix}^{-1} = \\frac{1}{\\det A} \\operatorname {adj}A \\displaystyle = \\frac{1}{aei+bfg+cdh-afh-bdi-ceg}\\begin{bmatrix} ei-fh & ch-bi & bf-ce \\\\ fg-di & ai-cg & cd-af \\\\ dh-eg& bg-ah & ae-bd\\end{bmatrix}
2.3. 사차 정사각행렬 ✎ ⊖
- \\displaystyle A_{4\\times 4}^{-1}=\\begin{bmatrix}a&b&c&d \\\\ e&f&g&h \\\\ i&j&k&l \\\\ m&n&o&p \\end{bmatrix}^{-1} = \\frac{1}{\\det A} \\operatorname{ adj} A
=\\displaystyle\\frac{1}{a f k p + a g l n + a h j o + b e l o + b g i p + b h k m + c e j p + c f l m + c h i n + d e k n + d f i o + d g j m - a f l o - a g j p - a h k n - b e k p - b g l m - b h i o - c e l n - c f i p - c h j m - d e j o - d f k m - d g i n} \\begin{bmatrix}fkp+gln+hjo-flo-gjp-hkn&blo+cjp+dkn-bkp-cln-djo&bgp+chn+dfo-bho-cfp-dgn&bhk+cfl+dgj-bgl-chj-dfk \\\\ elo+gip+hkm-ekp-glm-hip&akp+clm+djo-alo-cjp-dkm&aho+cep+dgm-agp-chm-deo&agl+chi+dek-ahk-cel-dgi \\\\ ejp+flm+hin-eln-fip-hjm&aln+bip+djm-ajp-blm-din&afp+bhm+den-ahn-bep-dfm&ahj+bel+dfi-afl-bhi-dej \\\\ ekn+fio+gjm-ejo-fkm-gin&ajo+bkm+cin-akn-bio-cjm & agn+beo+cfm-afo-bgm-cen & afk + bgi + cej - agj - bek - cfi \\end{bmatrix}
2.4. 오차 정사각행렬 ✎ ⊖
- \\displaystyle A_{5\\times 5}^{-1}=\\begin{bmatrix}a&b&c&d&e \\\\ f&g&h&i&j \\\\ k&l&m&n&o \\\\ p&q&r&s&t \\\\ u&v&w&x&y \\end{bmatrix}^{-1} = \\frac{1}{\\det A} \\operatorname{ adj} A
=\\displaystyle\\frac{1}{agmsy+agntw+agorx+ahltx+ahnqy+ahosv+ailry+aimtv+aioqw+ajlsw+ajmqx+ajnrv+bfmtx+bfnry+bfosw+bhksy+bhntu+bhopx+biktw+bimpy+bioru+bjkrx+bjmsu+bjnpw+cflsy+cfntv+cfoqx+cgktx+cgnpy+cgosu+cikqy+ciltu+ciopv+cjksv+cjlpx+cjnqu+dfltw+dfmqy+dforv+dgkry+dgmtu+dgopw+dhktv+dhlpy+dhoqu+djkqw+djlru+djmpv+eflrx+efmsv+efnqw+egksw+egmpx+egnru+ehkqx+ehlsu+ehnpv+eikrv+eilpw+eimqu-agmtx-agnry-agosw-ahlsy-ahntv-ahoqx-ailtw-aimqy-aiorv-ajlrx-ajmsv-ajnqw-bfmsy-bfntw-bforx-bhktx-bhnpy-bhosu-bikry-bimtu-biopw-bjksw-bjmpx-bjnru-cfltx-cfnqy-cfosv-cgksy-cgntu-cgopx-ciktv-cilpy-cioqu-cjkqx-cjlsu-cjnpv-dflry-dfmtv-dfoqw-dgktw-dgmpy-dgoru-dhkqy-dhltu-dhopv-djkrv-djlpw-djmqu-eflsw-efmqx-efnrv-egkrx-egmsu-egnpw-ehksv-ehlpx-ehnqu-eikqw-eilru-eimpv} \\begin{bmatrix}gmsy+gntw+gorx+hltx+hnqy+hosv+ilry+imtv+ioqw+jlsw+jmqx+jnrv-gmtx-gnry-gosw-hlsy-hntv-hoqx-iltw-imqy-iorv-jlrx-jmsv-jnqw&bmtx+bnry+bosw+clsy+cntv+coqx+dltw+dmqy+dorv+elrx+emsv+enqw-bmsy-bntw-borx-cltx-cnqy-cosv-dlry-dmtv-doqw-elsw-emqx-enrv&bhsy+bitw+bjrx+cgtx+ciqy+cjsv+dgry+dhtv+djqw+egsw+ehqx+eirv-bhtx-biry-bjsw-cgsy-citv-cjqx-dgtw-dhqy-djrv-egrx-ehsv-eiqw&bhox+bimy+bjnw+cgny+ciov+cjlx+dgow+dhly+djmv+egmx+ehnv+eilw-bhny-biow-bjmx-cgox-cily-cjnv-dgmy-dhov-djlw-egnw-ehlx-eimv&bhnt+bior+bjms+cgos+cilt+cjnq+dgmt+dhoq+djlr+egnr+ehls+eimq-bhos-bimt-bjnr-cgnt-cioq-cjls-dgor-dhlt-djmq-egms-ehnq-eilr \\\\fmtx+fnry+fosw+hksy+hntu+hopx+iktw+impy+ioru+jkrx+jmsu+jnpw-fmsy-fntw-forx-hktx-hnpy-hosu-ikry-imtu-iopw-jksw-jmpx-jnru&amsy+antw+aorx+cktx+cnpy+cosu+dkry+dmtu+dopw+eksw+empx+enru-amtx-anry-aosw-cksy-cntu-copx-dktw-dmpy-doru-ekrx-emsu-enpw&ahtx+airy+ajsw+cfsy+citu+cjpx+dftw+dhpy+djru+efrx+ehsu+eipw-ahsy-aitw-ajrx-cftx-cipy-cjsu-dfry-dhtu-djpw-efsw-ehpx-eiru&ahny+aiow+ajmx+cfox+ciky+cjnu+dfmy+dhou+djkw+efnw+ehkx+eimu-ahox-aimy-ajnw-cfny-ciou-cjkx-dfow-dhky-djmu-efmx-ehnu-eikw&ahos+aimt+ajnr+cfnt+ciop+cjks+dfor+dhkt+djmp+efms+ehnp+eikr-ahnt-aior-ajms-cfos-cikt-cjnp-dfmt-dhop-djkr-efnr-ehks-eimp \\\\flsy+fntv+foqx+gktx+gnpy+gosu+ikqy+iltu+iopv+jksv+jlpx+jnqu-fltx-fnqy-fosv-gksy-gntu-gopx-iktv-ilpy-ioqu-jkqx-jlsu-jnpv&altx+anqy+aosv+bksy+bntu+bopx+dktv+dlpy+doqu+ekqx+elsu+enpv-alsy-antv-aoqx-bktx-bnpy-bosu-dkqy-dltu-dopv-eksv-elpx-enqu&agsy+aitv+ajqx+bftx+bipy+bjsu+dfqy+dgtu+djpv+efsv+egpx+eiqu-agtx-aiqy-ajsv-bfsy-bitu-bjpx-dftv-dgpy-djqu-efqx-egsu-eipv&agox+aily+ajnv+bfny+biou+bjkx+dfov+dgky+djlu+eflx+egnu+eikv-agny-aiov-ajlx-bfox-biky-bjnu-dfly-dgou-djkv-efnv-egkx-eilu&agnt+aioq+ajls+bfos+bikt+bjnp+dflt+dgop+djkq+efnq+egks+eilp-agos-ailt-ajnq-bfnt-biop-bjks-dfoq-dgkt-djlp-efls-egnp-eikq \\\\fltw+fmqy+forv+gkry+gmtu+gopw+hktv+hlpy+hoqu+jkqw+jlru+jmpv-flry-fmtv-foqw-gktw-gmpy-goru-hkqy-hltu-hopv-jkrv-jlpw-jmqu&alry+amtv+aoqw+bktw+bmpy+boru+ckqy+cltu+copv+ekrv+elpw+emqu-altw-amqy-aorv-bkry-bmtu-bopw-cktv-clpy-coqu-ekqw-elru-empv&agtw+ahqy+ajrv+bfry+bhtu+bjpw+cftv+cgpy+cjqu+efqw+egru+ehpv-agry-ahtv-ajqw-bftw-bhpy-bjru-cfqy-cgtu-cjpv-efrv-egpw-ehqu&agmy+ahov+ajlw+bfow+bhky+bjmu+cfly+cgou+cjkv+efmv+egkw+ehlu-agow-ahly-ajmv-bfmy-bhou-bjkw-cfov-cgky-cjlu-eflw-egmu-ehkv&agor+ahlt+ajmq+bfmt+bhop+bjkr+cfoq+cgkt+cjlp+eflr+egmp+ehkq-agmt-ahoq-ajlr-bfor-bhkt-bjmp-cflt-cgop-cjkq-efmq-egkr-ehlp \\\\flrx+fmsv+fnqw+gksw+gmpx+gnru+hkqx+hlsu+hnpv+ikrv+ilpw+imqu-flsw-fmqx-fnrv-gkrx-gmsu-gnpw-hksv-hlpx-hnqu-ikqw-ilru-impv&alsw+amqx+anrv+bkrx+bmsu+bnpw+cksv+clpx+cnqu+dkqw+dlru+dmpv-alrx-amsv-anqw-bksw-bmpx-bnru-ckqx-clsu-cnpv-dkrv-dlpw-dmqu&agrx+ahsv+aiqw+bfsw+bhpx+biru+cfqx+cgsu+cipv+dfrv+dgpw+dhqu-agsw-ahqx-airv-bfrx-bhsu-bipw-cfsv-cgpx-ciqu-dfqw-dgru-dhpv&agnw+ahlx+aimv+bfmx+bhnu+bikw+cfnv+cgkx+cilu+dflw+dgmu+dhkv-agmx-ahnv-ailw-bfnw-bhkx-bimu-cflx-cgnu-cikv-dfmv-dgkw-dhlu&agms+ahnq+ailr+bfnr+bhks+bimp+cfls+cgnp+cikq+dfmq+dgkr+dhlp-agnr-ahls-aimq-bfms-bhnp-bikr-cfnq-cgks-cilp-dflr-dgmp-dhkq\\end{bmatrix}.
3. 참고 문헌 ✎ ⊖
- Horn, Roger A.; Johnson, Charles R. (2013), Matrix Analysis (2nd ed.), Cambridge University Press, ISBN 978-0-521-54823-6
