Matrices thiab determinants tau pom nyob rau hauv lub xyoo pua yim thiab kaum cuaj. Thaum pib, lawv txoj kev loj hlob cuam tshuam txog kev hloov pauv ntawm cov khoom geometric thiab kev daws teeb meem ntawm cov kab sib npaug ntawm cov kab sib npaug. Keeb kwm, qhov tseem ceeb thaum ntxov yog ntawm tus txiav txim. Nyob rau hauv niaj hnub linear algebra ua txoj kev, matrices yog xam thawj. Nws tsim nyog xav txog lo lus nug no ib pliag.
Cov lus teb los ntawm cheeb tsam kev paub
Matrices muab txoj hauv kev thiab siv tau zoo los daws ntau yam teeb meem, xws li:
- systems of linear equations;
- kev sib npaug ntawm cov khoom (hauv physics);
- graph theory;
- Leontief tus qauv kev lag luam;
- txoj hav zoov;
- khoos phis tawj duab thiab tomography;
- genetics;
- cryptography;
- hluav taws xob tes hauj lwm;
- fractal.
Qhov tseeb, matrix algebra rau "dummies" muaj qhov yooj yim txhais. Nws yog qhia raws li nram no: qhov no yog ib tug scientific teb ntawm kev paub nyob rau hauv uascov txiaj ntsig hauv nqe lus nug yog kawm, txheeb xyuas thiab tshawb nrhiav tag nrho. Hauv seem no ntawm algebra, ntau yam haujlwm ntawm matrices hauv kev kawm tau kawm.
Yuav ua li cas nrog matrices
Cov txiaj ntsig no suav tias yog sib npaug yog tias lawv muaj qhov sib npaug thiab txhua lub ntsiab lus ntawm ib qho sib npaug rau cov khoom sib txuas ntawm lwm tus. Nws yog ua tau kom muab ib tug matrix los ntawm ib tug tas li. Qhov muab no hu ua scalar multiplication. Piv txwv: 2=[1234]=[2⋅12⋅32⋅22⋅4]=[2468].
Matrices ntawm tib qhov loj tuaj yeem ntxiv thiab rho tawm los ntawm cov khoom siv, thiab cov txiaj ntsig ntawm qhov ntau thiab tsawg tuaj yeem sib npaug. Piv txwv li: ntxiv ob A thiab B: A=[21−10]B=[1423]. Qhov no tuaj yeem ua tau vim A thiab B yog ob qho tib si matrices nrog ob kab thiab tib tus lej ntawm kab. Nws yog ib qho tsim nyog ntxiv txhua lub ntsiab lus hauv A rau cov khoom sib xws hauv B: A + B=[2 + 11 + 2− 1 + 40 + 3]=[3333]. Matrices raug rho tawm tib yam hauv algebra.
Matrix sib npaug ua haujlwm sib txawv me ntsis. Ntxiv mus, tuaj yeem muaj ntau qhov xwm txheej thiab kev xaiv, nrog rau cov kev daws teeb meem. Yog tias peb muab cov matrix Apq thiab Bmn, ces cov khoom Ap × q + Bm × n=[AB] p × n. Kev nkag mus rau hauv kab gth thiab kab hth ntawm AB yog cov lej ntawm cov khoom ntawm cov khoom nkag hauv g A thiab h B. Nws tsuas yog ua tau kom muab ob lub matrices yog tias tus naj npawb ntawm kab hauv thawj thiab kab hauv qhov thib ob. sib npaug. Piv txwv li: ua kom tiav qhov kev txiav txim siab A thiab B: A=[1−130]B=[2−11214]. Qhov no ua tau vim tias thawj matrix muaj 2 kab thiab qhov thib ob muaj 2 kab. AB=[1⋅2+3⋅−1−1⋅2+0⋅−11⋅1+3⋅2−1⋅1+0⋅21⋅1+3⋅4−1⋅1+0⋅4]=[−1−27–113–1].
Cov ntaub ntawv tseem ceeb ntawm matrices
Qhov tseem ceeb hauv nqe lus nug npaj cov ntaub ntawv xws li qhov sib txawv thiab qhov tsis tu ncua thiab khaws cia rau hauv kab thiab kab, feem ntau hu ua C. Txhua txoj haujlwm hauv matrix yog hu ua lub caij. Piv txwv li: C=[1234]. Nws muaj ob kab thiab ob kab. Element 4 yog nyob rau hauv kab 2 thiab kem 2. Feem ntau koj tuaj yeem sau lub npe matrix tom qab nws qhov ntev, lub npe Cmk muaj m kab thiab k kab.
Expanded matrices
Kev txiav txim siab yog qhov muaj txiaj ntsig zoo kawg uas tuaj txog hauv ntau qhov chaw thov sib txawv. Matrices yog Ameslikas los ntawm cov kab ke ntawm cov kab zauv sib npaug. Muab cov qauv hauv qab no ntawm kev tsis sib xws, cov hauv qab no ua tiav matrix yuav tsum tau muab rau hauv tus account:
2x + 3y – z=6
–x – y – z=9
x + y + 6z=0.
Sau cov coefficients thiab cov lus teb qhov tseem ceeb, suav nrog tag nrho cov cim rho tawm. Yog hais tias lub caij nrog tus lej tsis zoo, ces nws yuav muab sib npaug rau "1". Ntawd yog, muab cov kab ke ntawm (linear) sib npaug, nws tuaj yeem koom nrog matrix (daim phiaj ntawm cov lej hauv cov lej) nrog nws. Nws yog ib qho uas tsuas muaj cov coefficients ntawm linear system. Qhov no hu ua "expanded matrix". Daim phiaj uas muaj cov coefficients ntawm sab laug ntawm txhua qhov sib npaug tau "padded" nrog cov lus teb los ntawm sab xis ntawm txhua qhov sib npaug.
Cov ntaub ntawv, uas yogB qhov tseem ceeb ntawm matrix sib haum rau x-, y-, thiab z qhov tseem ceeb hauv qhov system qub. Yog hais tias nws yog npaj kom raug, ces ua ntej ntawm tag nrho cov xyuas nws. Qee lub sij hawm koj yuav tsum tau rov kho cov ntsiab lus lossis ntxig xoom ua qhov chaw nyob hauv matrix tau kawm lossis kawm.
Muab cov kab ke hauv qab no ntawm kev sib npaug, peb tuaj yeem sau tam sim ntawd qhov cuam tshuam augmented matrix:
x + y=0
y + z=3
z – x=2.
Ua ntej, nco ntsoov rov kho qhov system li:
x + y=0
y + z=3
–x + z=2.
Tom qab ntawd nws tuaj yeem sau cov matrix txuam xws li: [11000113-1012]. Thaum tsim ib qho txuas ntxiv, nws tsim nyog siv xoom rau txhua cov ntaub ntawv uas qhov chaw sib thooj hauv qhov system ntawm kab sib npaug yog khoob.
Matrix Algebra: Cov Khoom Ua Haujlwm
Yog tias tsim nyog los tsim cov ntsiab lus tsuas yog los ntawm tus nqi coefficient, ces tus nqi txiav txim siab yuav zoo li no: [110011-101]. Qhov no hu ua "coefficient matrix".
Muab rau hauv tus account hauv qab no txuas ntxiv matrix algebra, nws yog qhov tsim nyog los txhim kho nws thiab ntxiv cov kab sib txuas. Uas tau hais tias, nws yog ib qho tseem ceeb uas yuav tsum nco ntsoov tias lawv xav tau cov kev hloov pauv kom zoo thiab zoo. Thiab feem ntau thaum muaj peb qhov sib txawv, siv x, y thiab z hauv qhov kev txiav txim ntawd. Yog li ntawd, cov kab ke sib txuas yuav tsum yog:
x + 3y=4
2y - z=5
3x + z=-2.
Matrix loj
Cov khoom hauv nqe lus nug feem ntau raug xa mus los ntawm lawv qhov kev ua tau zoo. Qhov loj ntawm lub matrix hauv algebra yog muab raws likev ntsuas, txij li chav tsev tuaj yeem hu ua txawv. Kev ntsuas ntsuas ntawm qhov tseem ceeb yog kab thiab kab, tsis yog qhov dav thiab ntev. Piv txwv li, matrix A:
[1234]
[2345]
[3456].
Vim A muaj peb kab thiab plaub kab, qhov loj ntawm A yog 3 × 4.
→
↓
Kab mus rau sab. Cov kab nce thiab nqis. "Law" thiab "kem" yog specifications thiab tsis interchangeable. Matrix qhov ntau thiab tsawg yog ib txwm teev nrog tus lej ntawm kab thiab tom qab ntawd tus lej ntawm kab. Ua raws li lub rooj sib tham no, cov hauv qab no B:
[123]
[234] yog 2 × 3. Yog hais tias ib tug matrix muaj tib cov kab raws li kab, ces nws yog hu ua "square". Piv txwv li, tus nqi coefficient los ntawm saum toj no:
[110]
[011]
[-101] yog 3 × 3 square matrix.
Matrix sau thiab formatting
Formatting note: Piv txwv li, thaum koj xav sau ib lub matrix, nws yog ib qho tseem ceeb uas yuav tau siv brackets . Tus nqi tsis muaj txiaj ntsig || tsis siv vim tias lawv muaj qhov sib txawv ntawm cov ntsiab lus no. Cov kab lus lossis curly braces {} yeej tsis siv. Los yog qee qhov lwm pawg cim, lossis tsis muaj dab tsi, vim tias cov kev nthuav qhia no tsis muaj lub ntsiab lus. Nyob rau hauv algebra, ib tug matrix yog ib txwm nyob rau hauv square brackets. Tsuas yog cov ntawv sau raug yuav tsum raug siv, lossis cov lus teb yuav raug suav tias yog garbled.
Raws li tau hais ua ntej, cov txiaj ntsig muaj nyob hauv matrix yog hu ua cov ntaub ntawv. Txawm li cas los xij, cov ntsiab lus hauv nqe lus nug feem ntau yog sauCov tsiaj ntawv loj, xws li A lossis B, thiab cov ntawv sau tau teev tseg siv cov tsiaj ntawv me, tab sis nrog cov ntawv sau npe. Hauv matrix A, cov nqi feem ntau hu ua "ai, j", qhov twg kuv yog kab ntawm A thiab j yog kab ntawm A. Piv txwv li, a3, 2=8. Qhov nkag rau a1, 3 yog 3.
Rau cov matrices me, cov uas muaj tsawg tshaj li kaum kab thiab kab, cov ntawv sau npe yog qee zaum raug tshem tawm. Piv txwv li, "a1, 3=3" tuaj yeem sau ua "a13=3". Pom tseeb qhov no yuav tsis ua haujlwm rau cov matrices loj vim a213 yuav tsis meej.
Matrix yam
Qee zaum cais raws li lawv cov ntaub ntawv teev tseg. Piv txwv li, xws li ib tug matrix uas muaj tag nrho cov xoom nkag hauv qab kab pheeb ces kaum sab saum toj-sab laug-hauv qab-txoj cai "diagonal" yog hu ua sab sauv daim duab peb sab. Ntawm lwm yam, tej zaum yuav muaj lwm yam thiab hom, tab sis lawv tsis muaj txiaj ntsig zoo. Feem ntau, feem ntau pom tau tias yog daim duab peb sab qaum. Cov nqi uas tsis yog xoom exponents horizontally hu ua kab pheeb ces kaum. Cov hom zoo sib xws muaj cov ntawv tsis yog xoom uas tag nrho yog 1, cov lus teb zoo li no yog hu ua zoo tib yam (rau cov laj thawj uas yuav paub meej thaum nws kawm thiab nkag siab yuav ua li cas muab cov txiaj ntsig hauv nqe lus nug). Muaj ntau qhov kev tshawb fawb zoo sib xws. Lub 3 × 3 tus kheej yog qhia los ntawm I3. Ib yam li ntawd, 4 × 4 tus kheej yog I4.
Matrix Algebra thiab Linear Spaces
Nco ntsoov tias daim duab peb sab matrices yog square. Tab sis cov diagonals yog triangular. Nyob rau hauv view ntawm no, lawv yogsquare. Thiab cov cim yog suav tias yog kab pheeb ces kaum thiab, yog li ntawd, daim duab peb sab thiab square. Thaum nws yuav tsum tau piav qhia txog matrix, ib qho feem ntau tsuas yog qhia ib tus kheej cov kev faib tawm tshwj xeeb tshaj plaws, vim qhov no txhais tau tias tag nrho lwm tus. Muab faib cov kev tshawb fawb hauv qab no:li 3 × 4. Qhov no, lawv tsis yog square. Yog li ntawd, cov txiaj ntsig tsis tuaj yeem yog lwm yam. Cov kev faib tawm hauv qab no:yog ua tau li 3 × 3. Tab sis nws suav hais tias yog square, thiab tsis muaj dab tsi tshwj xeeb txog nws. Kev faib tawm ntawm cov ntaub ntawv hauv qab no:raws li 3 × 3 sab sauv daim duab peb sab, tab sis nws tsis yog kab pheeb ces kaum. Muaj tseeb, nyob rau hauv qhov tseem ceeb nyob rau hauv kev txiav txim siab tej zaum yuav muaj zeros ntxiv rau lossis siab dua qhov chaw nyob thiab qhia qhov chaw. Kev faib tawm raws li kev kawm yog ntxiv: [0 0 1] [1 0 0] [0 1 0], qhov twg nws yog sawv cev raws li kab pheeb ces kaum thiab, ntxiv rau, cov ntawv nkag yog tag nrho 1. Ces qhov no yog 3 × 3 tus kheej., I3.
Vim hais tias cov matrices analogous yog los ntawm lub ntsiab lus square, koj tsuas yog yuav tsum tau siv ib qho kev ntsuas kom pom lawv qhov ntev. Rau ob lub matrices kom sib npaug, lawv yuav tsum muaj tib qhov parameter thiab muaj tib qhov nkag hauv tib qhov chaw. Piv txwv li, xav tias muaj ob lub ntsiab lus hauv qab no: A=[1 3 0] [-2 0 0] thiab B=[1 3] [-2 0]. Cov nqi no yuav tsis zoo ib yam li qhov sib txawv ntawm qhov loj.
Txawm tias A thiab B yog: A=[3 6] [2 5] [1 4] thiab B=[1 2 3] [4 5 6] - lawv tseem tsis tau zoo ib yam tib yam. A thiab B txhua tus muajrau qhov nkag thiab kuj muaj tib tus lej, tab sis qhov no tsis txaus rau matrices. A yog 3 × 2. Thiab B yog 2 × 3 matrix A rau 3 × 2 tsis yog 2 × 3. Nws tsis muaj teeb meem yog A thiab B muaj cov ntaub ntawv tib yam lossis cov lej tib yam li cov ntaub ntawv. Yog tias A thiab B tsis zoo ib yam loj thiab cov duab, tab sis muaj qhov sib npaug ntawm qhov chaw zoo sib xws, lawv tsis sib npaug.
Kev ua haujlwm zoo ib yam hauv cheeb tsam hauv kev txiav txim siab
Cov cuab yeej ntawm matrix vaj huam sib luag no tuaj yeem hloov mus ua haujlwm rau kev tshawb fawb ywj pheej. Piv txwv li, ob lub matrices tau muab, thiab nws tau qhia tias lawv sib npaug. Hauv qhov no, koj yuav tsum tau siv qhov sib npaug no los tshawb nrhiav thiab tau txais cov lus teb rau cov txiaj ntsig ntawm qhov sib txawv.
Piv txwv thiab cov kev daws teeb meem ntawm matrices hauv algebra tuaj yeem sib txawv, tshwj xeeb tshaj yog thaum nws los txog rau qhov sib npaug. Muab hais tias cov matrices hauv qab no raug txiav txim siab, nws yog ib qho tsim nyog yuav tsum nrhiav qhov x thiab y qhov tseem ceeb. Rau A thiab B kom sib npaug, lawv yuav tsum yog qhov loj thiab zoo tib yam. Qhov tseeb, lawv yog xws li, vim hais tias txhua tus ntawm lawv yog 2 × 2 matrices. Thiab lawv yuav tsum muaj qhov tseem ceeb ntawm tib qhov chaw. Tom qab ntawd a1, 1 yuav tsum sib npaug b1, 1, a1, 2 yuav tsum sib npaug b1, 2, thiab lwm yam). Tab sis, a1, 1=1 yog pom tseeb tsis sib npaug rau b1, 1=x. Rau A kom zoo ib yam rau B, qhov nkag yuav tsum muaj a1, 1=b1, 1, yog li nws muaj peev xwm ua 1=x. Ib yam li ntawd, qhov ntsuas a2, 2=b2, 2, yog li 4=y. Ces tus tov yog: x=1, y=4. Muab hais tias cov nram qab nomatrices sib npaug, koj yuav tsum nrhiav qhov tseem ceeb ntawm x, y thiab z. Yuav kom muaj A=B, cov coefficients yuav tsum muaj tag nrho cov khoom sib npaug. Ntawd yog, a1, 1=b1, 1, a1, 2=b1, 2, a2, 1=b2, 1 thiab lwm yam. Tshwj xeeb, yuav tsum:
4=x
-2=y + 4
3=z / 3.
Raws li koj tuaj yeem pom los ntawm cov lej xaiv: nrog 1, 1-, 2, 2- thiab 3, 1- ntsiab lus. Kev daws peb qhov kev sib npaug no, peb tau txais cov lus teb: x=4, y=-6 thiab z=9. Matrix algebra thiab matrix kev ua haujlwm txawv ntawm txhua tus neeg siv los, tab sis lawv tsis tuaj yeem tsim dua.
xov xwm ntxiv hauv cheeb tsam no
Linear matrix algebra yog qhov kev kawm ntawm cov khoom sib npaug sib npaug thiab lawv cov khoom hloov pauv. Txoj kev paub no tso cai rau koj los txheeb xyuas cov kev sib hloov hauv qhov chaw, kwv yees tsawg kawg squares, daws cov kev sib npaug sib txawv, txiav txim siab lub voj voog hla peb lub ntsiab lus, thiab daws ntau yam teeb meem hauv lej, physics thiab thev naus laus zis. Lub linear algebra ntawm matrix tsis yog qhov kev nkag siab ntawm lo lus siv, uas yog, qhov chaw vector v dhau ib daim teb f, thiab lwm yam.
Matrix thiab determinant yog cov cuab yeej linear algebra zoo heev. Ib qho ntawm cov haujlwm tseem ceeb yog kev daws teeb meem ntawm matrix kab zauv Ax=b, rau x. Txawm hais tias qhov no tuaj yeem daws qhov kev xav tau los ntawm kev siv tus inverse x=A-1 b. Lwm txoj hauv kev, xws li kev tshem tawm Gaussian, yog cov lej ntau txhim khu kev qha.
Ntxiv rau kev siv los piav qhia txog kev kawm ntawm cov kab sib npaug ntawm cov kab sib npaug, cov lus teev tseglo lus saum toj no kuj yog siv los piav txog ib hom algebra. Tshwj xeeb tshaj yog, L hla ib daim teb F muaj cov qauv ntawm lub nplhaib nrog tag nrho cov axioms ib txwm muaj rau kev sib ntxiv thiab kev sib npaug, ua ke nrog cov kev cai sib faib. Yog li ntawd, nws muab nws cov qauv ntau dua li lub nplhaib. Linear matrix algebra kuj lees paub qhov kev ua haujlwm sab nrauv ntawm kev sib faib los ntawm scalars uas yog cov ntsiab lus ntawm qhov chaw hauv qab F. Piv txwv li, cov txheej txheem ntawm txhua qhov kev txiav txim siab hloov pauv ntawm qhov chaw vector V mus rau nws tus kheej dhau ntawm daim teb F yog tsim dua F. Lwm qhov piv txwv ntawm linear algebra yog cov txheej ntawm tag nrho cov square matrices tshaj ib daim teb R cov lej tiag.
