LU Decomposition | MIT 18.06SC Linear Algebra, Fall 2011

[@kevinwu5954]

I love MIT Open Courseware. They always allow me to understand theorems faster!!

[@stopasking4962]

DUDE! I Had this instructor! HE IS AMAZING!

[@yuchenliu4695]

This video literally solves the problem that has haunted me for weeks. He is my life saver.

[@coolkidz4lyf]

dude you were great...thanks a million

[@georgesadler7830]

Professor Ben Harris, thank you for an excellent video/lecture on LU Decomposition in Linear Algebra. This is an error free video/lecture on KZbin.

[@NegativeNezzy]

This was amazingly helpful, thank you very much. Congratulations on keeping such meticulous records of your steps.

[@Eyenn_n]

Very well explained. Thank you very much!

[@arestes]

the elementary matrix needed for permutation is not lower triangular so it's not useful for getting our L matrix since it would defeat the purpose. The best you can do is getting an extra permutation matrix known as P to get PA=LU but you can't be guaranteed to get A=LU by exchanging rows.

[@ismailcakr9086]

better than my teacher even i dont speak english lol

[@freddoran1665]

İtüden geldik biz de buraya :( Mat281'in ABV

[@kdoodoo]

better than my teacher even he speaks english lol

[@greg0r-32]

Learnt the method quicker than in 2 hours at my numerical methods class. Superb.

[@MrRenanwill]

Yes! Thanks for your comment at end of the video! I was requiring that U was not singular, but it's not necessary for LU decomposition. =) So... if you want to ensure that the LU decomposition exists you just need all submatrixs be nonsingular except the matrix you are trying to decompose in LU.

[@DaisyZhangAI]

Yup, 10 minutes solved my week-long agony.

[@유제환-l3n]

Thank you Ben

[@7556wjq]

I finally understood this!

[@MrTrollmojo]

Thank you verry much Ben :D , I ve learned so much!!!

[@alexconstandache907]

Well done presentation/explanation of LU Decomposition. Thank you for that Ben. :)

[@sheyhan1]

Thanx to you now I know LU factorization. Thank you sir. you are doing a great job.

[@orkhanmd]

Great explanation. Thank you, man!

[@Lilg4ever2396]

your a life saver ben

[@MrHarpreet99]

Very nice explanation, Ben, Understood each and everything of it. Thank you very much.

[@RecklezzMusic]

Very clear and concise. Thanks for the example

[@carolinetleane6697]

much appreciation! thanks Ben

[@nanakojonyarko303]

good work done Ben keep it up

[@nguyenthanhdat93]

Thanks for sharing. Wonderful explanation!!

[@TheUmangTarang]

Just missed one point here: the decomposition will exists if a=0 and b=0 at the same time.

[@obi-wankenobi9871]

In tasks like that one, you usually have a restriced domain, so you arent allowed to simply use zero.

[@xploi]

You'll need to use the permutation matrix known as P, so PA = LU. Google it, cheers.

[@ProudQLD]

Very comprehensive. Thank you.

[@vitoralves9850]

AMAZING

[@abrarshariare5835]

thanks a million dude

[@silencedidgood]

Ben.....you're doing a great job but you guys should always pay homage to Gilbert Strang as well.....Thanks for your most excellent factorization.

[@mehmetaliozer2403]

nice explanation thanks 👍👍

[@erkanturan7215]

this is very helpful!

[@sahamathaque5088]

I dont get how he worked out the elimination matrix? at 2:35 can someone help please.

[@ElektrikAkar]

+Sahamat Haque Think about rows. Look at the indices of elements in E matrix. E matrix is like; e11 e12 e13 e21 e22 e23 e31 e32 e33 and an arbitrary element is eij isn't it? Here i corresponds NEW matrix's row and the j is old matrix's row. Let me explain; e11 = 1st row of NEW matrix includes 1st row of old matrix how many times? Of course 1 time. Because they are idendical. Old 1st row * 1 = New 1st row. e12 = 1st row of NEW matrix includes 2nd row of old matrix how many times? ZERO. e13 = also zero. But for the elimination pusposes what he did is that he added first row of old matrix to second row. To eliminate something. How many times? a times. So new 2nd row is = OLD's second row + a times OLD's first row. e21 = 2nd row of NEW includes how many of 1st row of OLD = a ; e22 = 2nd row of NEW includes one times of 2nd row of OLD = 1; e23 = we did not do anything about 3rd row so is 0. It is like that.

[@sahamathaque5088]

thank you very much! This was very helpful :D

[@yifeipeng4689]

Thank you so much

[@islozlemorunc3856]

Excellent!!! Thank you very much

[@hamishforrest7553]

Great explanation, thanks!

[@leilaineoshiro2958]

Awesome video!

[@lee_land_y69]

Why lu decomposition doesn’t work if we have to do row exchanges?

[@drdale104]

It can work, but then you have to multiply by a pivot matrix on both sides. To make things easier, if I'm in the middle of a problem and I notice I need to make a pivot I will restart the question and have my first step being multiplying by a pivot matrix. But you can usually tell right off the bat if you need to pivot. But for this problem, since it was working with variables, an assumption had to be made.

[@AmodSandeepa]

Thanks a lot.

[@seanjoe551]

Thanks a lot. It really helped me

[@riyadshauk2432]

Very helpful! Question: Why does LU Decomposition only work when not doing row exchanges to get U (9:15)?

[@Belandbec]

Because you are "registering" your operations in the L matrix. So if you change the order of the rows or columns, you may screw up with the L matrix elements position. That's my guess

[@FarazMazhar]

Damn, the dude solved it like a magic trick.

[@xploi]

Which university? Not MIT of course, this is just a 9min video, MIT has uploaded entire courses of linear algebra that you can take and understand these procedures.

[@punksnotdead4766]

VERY CLEAR, THANKS

[@anujkumar-cq5hx]

Cool and clear.

[@prakashkhadka6839]

Thank you sooo much sir

[@DMccloudy]

So clear ;A; Thanks!

[@uncannyvalley2084]

Thanks redpussy to make me understand how to read permutation matrix :D cause Gulina only could talk about donuts

[@VivekRoy2991]

Thanks a lot!

[@xploi]

I know, but that's because you have only watched this video. You can watch the whole course provided free by MIT to understand it.

[@DavidLee-dc1xe]

A little question: What if a and b both equal zero? I think in this case, the matrix has its LU-decomposition too.

[@coal2710]

No.We are assuming these matrices are elementary.And as a rule,if a row is full of zeros,that row should be the last row of matrix.

[@nadirarfi2342]

does this work if we put the ones in the U matrix instead of L

[@anoopk2922]

very helpful. thank you

[@raffaelenapoli2012]

in the U matrix if a=b the matrix U is singolar (det=o). a not ugual b is another condition for the U existence?

[@MrRenanwill]

No... U read the definition of U and you will see that U dont need to be nonsingular (... it's just a Upper triangular matrix. =)

[@nanakojonyarko303]

can I kindly get a reference material on this topic

[@凛-r3h]

what is L and what is U? why is it called that

[@carmendiazescribano7685]

thank you :)

[@lucaslimacosta5918]

Thanks!

[@embedlab]

Good One.. thumbs up MIT.... (Y)

[@davidlanning2]

Can someone explain me in more details the last thing he says. Why a-b (even though is a pivot) can be zero?

[@jogaserbia]

Salvatore Cipolla "a" is the pivot, not "a-b". at 3:56

[@fugisawa]

+jogaserbia +Salvatore Cipolla But a-b is also a pivot. It is the pivot of the third row, after the elimination. I believe (a - b) ≠ 0 is also a condition.

[@SilverArro]

+Daniel Fugisawa He explains this exactly at the end of the video. Row 3 does not need to have a pivot in order for A to successfully factor into LU. If the third column of A has no pivot, then it simply we means we have a singular, rank 2 matrix, and as Ben explains, singular matrices can still have LU decompositions. The important thing is to avoid row permutations, and since row 3 is the last row, there's no need to exchange it with another in order to transform A into U. From Wikipedia: "If A is a singular matrix of rank k, then it admits an LU factorization if the first k-leading principal minors are nonsingular." In the case where a=b, the matrix has rank 2 and the first 2 principal minors are 1 and a. Both of those are non-zero, and thus the LU decomposition still holds. Therefore, a - b CAN be zero.

[@Kangnkw]

Hi, I may be wrong, But, for upper triangular matrix, we are only concerned with entries below the diagonal ones to be zero. A Zero matrix (all zero elements-including diagonal) is still considered to be of a upper triangular form.

[@Zseselja]

Nice Job!!

[@danielfaraday4140]

Yellow head knows something.

[@jimmonte9826]

Suppose b=0. Then a0 is OK. So a0 alone is too restrictive.

[@jimmonte9826]

Sorry -- Meant to write that if b = 0, then a = 0 is allowed. L is the identity matrix and u11 = u13 = 1 and other uij are 0.

[@kellyfreet351]

"good"

[@batfishh]

you're god damn right

[@obi-wankenobi9871]

I now understand why people are impressed, when you say you were at the MIT. The explaination is garbage.

[@arestes]

seriously? you're watching a homework solution without knowing the theory? the complete lectures where the whole theory is developed and easy to understand is uploaded on MIT's channel. You can't just expect a whole lecture done from scratch for every homework solution posted.

[@paradisloola3336]

helpfull thanks.

[@roxasthegreek]

*Good.*

[@VanTran-rq3vz]

why couldn't we do row exchange in the elimination part in LU decomposition? I remember my professor taught about partial pivoting in Gauss Elimination and LU decomposition.

[@gabrielpereiramendes3463]

#Excelent!

[@ThisByGustav]

"Okey! Thanks."

[@kabirbaghel8835]

thenks

[@RomiiLeeh]

what does it mean "LU"?! :)

[@julianabucher]

L: lower triangular matrix. U: upper triangular matrix. This method factors A into two matrices, L * U.

[@RomiiLeeh]

Thank you so much! :)

[@neverbendorbreak]

Cool.

[@bastiaanbogers4114]

Okay... Thanks... :'D

[@danielkwon3062]

That is why its called MIT lol

[@bytekraken]

Great :D

[@speed15987]

you study this in MIT ? it's exactly like in our college :o I should study there :o

[@smithcodes1243]

Young Gilbert Strang haha

[@speed15987]

when you hear MIT, you expect some higher lvl things, but it turned out just like a normal college :/

[@Swizfisch]

weak ending

[@farzamimani5339]

daddy

[@32ia]

T U R K E y

[@skyrunners9082]

Hindi jante ho toh Bhagwan SIngh Vishwakarma ke channel se par lo.Faru samjhate hain.Ye firangi theek se para nahi pa raha hain

[@qnyazosmanov8980]

what a bad explanation blet, worse than even İbiş

[@kabirbaghel8835]

wat