Lej matrix. matrix npaug

Ntau ancient Suav lej siv nyob rau hauv lawv le caag post nyob rau hauv tabular daim ntawv nrog ib tug tej tus naj npawb ntawm kab thiab txhua. Ces, xws li zauv khoom xa mus rau raws li "khawv koob square". Txawm tias paub tus neeg mob ntawm txoj kev siv ntawm cov ntxhuav nyob rau hauv daim ntawv ntawm cov vajvoos peb sab, uas tsis tau lug txais yuav.

Rau hnub tim, ib tug zauv matrix kheev to taub obokt lub zoo nrog ib tug predetermined tooj ntawm txhua thiab cov cim uas txhais tau qhov ntev ntawm lub matrix. Nyob rau hauv kev kawm txog zauv, ib daim ntawv ntawm cov ntaubntawv povthawj siv tau lug siv rau cov ntaubntawv povthawj siv nyob rau hauv ib tug compact daim ntawv ntawm differential systems raws li zoo raws li ntawm linear algebraic equations. Nws yog assumed tias lub xov tooj ntawm kab nyob rau hauv lub matrix sib npaug zos rau cov xov tooj tam sim no nyob rau hauv lub system ntawm sib npaug, tus naj npawb ntawm cov kab sau raws nkaus Ii ntau npaum li cas tus tsis paub yuav tsum tau sau tseg nyob rau hauv lub chav kawm ntawm cov tshuaj.

Dhau li ntawm tus fact tias cov matrix nws tus kheej nyob rau hauv lub chav kawm ntawm nws cov tshuaj ua rau kev nrhiav cov tsis paub xam qhovkev nyob rau hauv lub zwj ceeb ntawm lub cev, muaj ib tug xov tooj ntawm algebraic haujlwm uas muaj cai nqa tshaj ib tug muab xaam kwv. Qhov no daim ntawv muaj xws qhov sib ntxiv ntawm matrices muaj tib qhov loj me. Lub npaug ntawm matrices nrog tsim nyog qhov ntev (nws yog tau mus muab ib tug matrix nrog ib sab muaj ib tug xov tooj ntawm txhua sib npaug zos rau cov xov tooj ntawm kab ntawm cov matrix nyob rau lwm yam sab nrauv). Nws kuj tau tso cai rau muab ib tug matrix los ntawm ib tug vector, los yog lub caij los yog rau lub hauv paus ntiv nplhaib (txwv tsis pub scalar).

Xav lub matrix npaug yuav tsum tau zoo saib xyuas kom nruj me ntsis ua ntej tus naj npawb ntawm kab sib npaug zos rau cov xov tooj ntawm kab ntawm lub thib ob. Txwv tsis pub, qhov kev txiav txim ntawm lub matrix tsis txhais. Raws li txoj cai, uas cov matrix-matrix npaug, txhua lub caij nyob rau hauv lub tshiab array yog sib npaug rau cov sum ntawm cov khoom ntawm cov coj cov ntsiab ntawm cov kab ntawm cov thawj matrix ntsiab los ntawm lwm yam kab.

Rau clarity, cia peb xav txog ib qho piv txwv ntawm yuav ua li cas matrix npaug tshwm sim. Siv lub matrix A

Lub ob hlis ntuj 3 -2

3 4 0

-1 2 -2,

muab nws los ntawm cov matrix B

3 -2

1 0

4 -3.

Lub caij ntawm thawj kab ntawm cov thawj kem ntawm lub resulting matrix yog sib npaug zos rau 2 * 3 + 3 * 1 + (- 2) * 4. Raws li, nyob rau hauv thawj kab nyob rau hauv lub thib ob kem caij yuav sib npaug 2 * (- 2) + 3 * 0 + (- 2) * (- 3), thiab thiaj li nyob kom txog rau thaum sau ntawm txhua lub caij ntawm lub tshiab matrix. Txoj Cai matrix npaug yuav hais tias cov kev tshwm sim ntawm ib yam khoom mxn matrix tsis los ntawm lub matrix muaj ib tug piv nxk, yuav muaj ib lub rooj uas muaj ib tug loj ntawm m x k. Tom qab txoj cai no, peb yuav xaus uas lub khoom ntawm lub thiaj li hu ua square matrices, ntsig txog, ntawm qhov kev txiav txim yog yeej ib txwm hais tseg.

Los ntawm cov khoom muaj los ntawm matrix npaug yuav tsum muab sau ua ke raws li ib tug yooj yim fact tias lub lag luam no tsis yog commutative. Qhov ntawd yog qhov khoom ntawm tus matrix M rau N yog tsis sib npaug zos rau cov khoom ntawm N los ntawm M. Yog hais tias nyob rau hauv square matrices ntawm qhov kev txiav txim yog pom tias lawv rau pem hauv ntej thiab rov qab cov khoom yog ib txwm txiav txim siab, differing xwb nyob rau hauv cov kev tshwm sim, lub lub matrix zoo li tej yam tej yam kev mob yog tsis ib txwm ua tiav raws li.

Nyob rau hauv matrix npaug muaj ib tug xov tooj ntawm thaj chaw uas muaj ib tug tseeb zauv proofs. Associativity txawj txhais tau tias fidelity nram qab no xyuam xim qhia: (MN) K = M (NK), qhov chaw uas M, N, thiab K - ib tug matrix muaj cov tsis uas npaug yog txhais. Distributivity npaug assumes uas M (N + K) = MN + MK, (M + N) K = MK + NK, L (MN) = (LM) N + M (haujlwm uas yog COy), qhov chaw uas L - xov tooj.

Qhov tsim nyog ntawm lub zog ntawm cov matrix npaug, hu ua lub "associative", nws li nram no tias nyob rau hauv ib yam khoom uas muaj nyob nruab nrab ntawm peb los yog ntau tshaj yam, pub nkag tsis siv nkhaus.

Siv cov distributive vaj tse muab lub sij hawm los mus qhia rau kev zawm hniav thaum xaiv cov kev matrix kab zauv. Thov nco ntsoov, yog hais tias peb qhib lub nkhaus, nws yog tsim nyog los khaws cia qhov kev txiav txim ntawm cov yam tseem ceeb.

Siv cov matrix kab zauv tsis tau tsuas yog compact cov ntaub ntawv cumbersome systems sib npaug, tab sis kuj tswj cov ua thiab cov kev daws.