This man single handedly saved my university algebra course, my teacher was just reading notes, he's actually expalining in a very clear manner.
@charmenk
11 жыл бұрын
Good professor with good old blackboard and white chalk teaching method. This is way better than all the fancy powerpoints that many teachers use now a days.
@9888565407
4 жыл бұрын
hey did ya benefit from these lectures
@yusufkavcakar8744
3 жыл бұрын
@@9888565407 yeah ı did
@jiayuanwang1498
2 жыл бұрын
I can't agree more!!!
@Tutkumsdream
11 жыл бұрын
Thanks to him! I passed Linear Algebra.. I watched his videos for 4 days before final exam and I got 74 from final.. If I couldnt watch Dr.Strang's lectures, I would probably fail...
@snoefnone9647
9 ай бұрын
For some reason i thought you were saying Dr. Strange's lecture!
@kayfouroneseven
12 жыл бұрын
this is a bazillion times more straightforward and clear than the lectures i pay for at my university. :( I appreciate this being online
@bsmichael9570
10 ай бұрын
He tells it like a story. It’s like he’s taking us all on a journey. You can’t wait to see the next episode.
@jollysan3228
9 жыл бұрын
I agree. > Just one small correction at 32:30: It should have been S * LAMBDA^100 * c instead of LAMBDA^100 * S * c.
@Slogan6418
5 жыл бұрын
thank you
@ozzyfromspace
4 жыл бұрын
The sad thing was, a few moments later he was struggling to explain things because even though he hadn't pinned down the error, he someone knew that something wasn't quite right. But he obviously had the core idea nailed
@alexandresoaresdasilva1966
4 жыл бұрын
thank you so much, was about to post asking about this.
@吴瀚宇
4 жыл бұрын
I stuck on this for like 10 mins, until I saw the comments here...
@maitreyverma2996
4 жыл бұрын
Perfect. I was about to write the same.
@apocalypse2004
8 жыл бұрын
I think Strang leaves out a key point in the difference equation example, which is that the n unique eigenvectors form a basis for R^n, which is why u0 can be expressed as a linear combination of the eigenvectors.
@alessapiolin
7 жыл бұрын
thanks!
@wontbenice
7 жыл бұрын
I was totally confused until you chimed in. Thx!
@seanmcqueen8498
6 жыл бұрын
Thank you for this comment!
@arsenron
6 жыл бұрын
in my opinion it is so obvious that it is not worth stopping on it
@dexterod
6 жыл бұрын
I think Strang assumed that A has n independent eigenvectors since most matrices do not have repeated eigenvalues.
@eye2eyeerigavo777
5 жыл бұрын
Math surprises you everytime...🤔 Never thought that connections between rate of growth in system dynamics, fibonacci Series and diagnalization of an INDEPENDENT vectors will finally boil down INTO GOLDEN RATIO OF EIGENVALUES at END! 😳
@cozmo4825
Жыл бұрын
Thank you a lot Prof. Strang you really made this course clear to understand, the way you teach these topics are superior, what really special about it is you put yourself in the position of your students answering their questions without them even asking, a true skill acquired not only by decades of teaching but actually having the mindset of a true teacher. Mr. Strang, words can't describe how good these lectures are its a true form of art. And thanks a lot for MIT OpenCourseWare for providing these lectures in good quality and free of charge, hopefully one day to I will pursue a master in my degree in electrical engineering in MIT.
@christoskettenis880
9 ай бұрын
The explanations of this professor of all those abstract theorems and blind methodologies are simply briliant
@albertacristie99
14 жыл бұрын
This is magnificiant!! I have no words to express how thankful I am towards the exposure of this video
@georgesadler7830
3 жыл бұрын
From this latest lecture , I am learning more about eigenvalues and eigenvectors in relation to diagonalization of a matrix. DR. Strang continues to increase my knowledge of linear algebra with these amazing lectures.
@BirnieMac1
7 ай бұрын
You know you’re in for some shenanigans when they pull out the “little trick” Professor Gilbert is an incredible teacher; I struggled with Eigenvalues and vectors in a previous course and this series of lectures has really helped understand it better Love your work Professor Gilbert
@ozzyfromspace
4 жыл бұрын
For the curious: F_100 = (a^99 - b^99) * b/sqrt(5) + a^99 , where a = (1 + sqrt(5))/2 and b = (1 - sqrt(5))/2 are the two eigenvalues of our system of difference equations. Numerically, F_100 = ~3.542248482 * 10^20 ... it's a very large number that grows like ~1.618^k 😲 Overall, great lecture Professor Strang! Thank you for posting, MIT OCW ☺️
@eroicawu
14 жыл бұрын
It's getting more and more interesting when differential equations are involved!
@SimmySimmy
5 жыл бұрын
through single matrix transformation, the whole subspace will expand or shrink with the rate of eigenvalues in the direction of its eigenvectors, suppose you can decompose a vector in this subspace into the linear combination of its eigenvectors, so after many times of the same transformation, the random vector will ultimately land on one of its eigenvectors with the largest eigenvalue.
@MsAlarman
2 жыл бұрын
Mindbending
@neoneo1503
3 жыл бұрын
A*S=S*Lambda (Using the linear combination view (Ax1=b1 column part) of Matrix Multiplication), That is Brilliant and Clear! Thanks!
@neoneo1503
3 жыл бұрын
Also expressing the state u_0 to u_k as linear combination of eigenvectors (at 30:00 and 50:00)
@Wabbelpaddel
3 жыл бұрын
Well, it's - if you interpret it that way - just a basis transformation from the standard base (up to isomorphism, then just additionally multiply the transforms of the alternate basis) onto the eigenvector basis. Provided of course, that either the characteristic polynomial factors distinctly, or that geometric and algebraic multiplicity match (because then the eigenspaces distinctly span the vector space up to isomorphism; if they weren't, you'd just have a subspace as a generating system). For anyone who wanted one more run-through.
@neoneo1503
3 жыл бұрын
@@Wabbelpaddel Thanks! =)
@cuinuc
14 жыл бұрын
I love professor Strang's great lectures. Just one small correction at 32:30: It should have been S * LAMBDA^100 * c instead of LAMBDA^100 * S * c.
@starriet
2 жыл бұрын
Nice catch!
@jeffery777
2 жыл бұрын
haha I think so
@eyuptarkengin816
8 ай бұрын
yeah, i though of the same thing and scrolled down the comments for a approval. Thanks mate :D
@sathviktummala5480
3 жыл бұрын
44:00 well that's an outstanding move
@kanikabagree1084
4 жыл бұрын
This teacher made fall in love with linear algebra thankyou ❤️
@syedsheheryarbokhari2780
Жыл бұрын
There is a small writing mistake at 32:30 by Prof Strang. He writes (eigenvalue matrix)^100 multiplying (eigenvector matrix) multiplying c's (constants). It ought to be (eigenvector matrix) multiplying (eigenvalue matrix)^100 multiplying c's. At the end of the lecture Professor Strang does narrate the correct formula but it is easier to miss.
@clutterbrainx
Жыл бұрын
Yeah I was confused for a very long time there
@rolandheinze7182
5 жыл бұрын
Hard lecture to get through personally but does illustrate some of the cool machinery for applying eigenvectors
@nguyenbaodung1603
3 жыл бұрын
I read something on SVD without even knowing about eigenvalues and eigenvectors, then watch a youtube video, explaining that V is actually the eigenvector decomposition of A^TA. Which is extremely insane when I got to see this video oh my godness. Now even haven't watched your SVD lecture, I can even tell the precise concept of it. Oh my godness Math is so perfect!!
@coreconceptclasses7494
4 жыл бұрын
I got 70 out of 75 in my final linear algebra exam thanks MIT...
@dalisabe62
4 жыл бұрын
The golden ratio arose from the Fibonacci sequence and has nothing to do with eigenvectors or eigenvalues. The beauty of using the eigenvectors and eigenvalue of a matrix though is limiting the effect of the transformation to the change in magnitude only, which reduces dynamics systems such as population growth that is a function of several variables to be encoded in a matrix computation without worrying about the effect of direction or rotation typically associated with matrix transformation. Since eigenvectors and eigenvalues change the magnitude of the parameter vector only, the idea of employing the Eigen transformation concept is quite genius. The same technique could be used in any dynamic system that could be modeled as a matrix transformation but one that produces a change in magnitude only.
@Arycke
11 ай бұрын
Hence the title of his *example* as "Fibonacci Example." Nowhere was it stated explicitly sthat the golden ratio didn't arise from the Fibonacci sequence, so I don't see where you got that from. The example has a lot to do with eigenvalues and eigenvectors by design, and is using a simple recurrence relation to show a use case. The Fibonacci sequence isn't unique anyway.
@go_all_in_777
6 ай бұрын
At 28:07, uk = (A^k)uo, can also be written as uk = S*(Lambda)^k*(S^-1)*uo. Also, we can write uo = S*c as explained at 30:00. therefore, uk = S*(Lambda)^k*(S^-1)*S*c=S*(Lambda)^k*c
@jeanpierre-st7rl
6 ай бұрын
Hi @ 29:46 Uo = C1X1 + C2X2 +C3X3... Is U0 a vector? If so, How can split this U0 in to a combination of eigen vectors? What is Ci ? If you have any info pleases let me know. Thanks.
@meetghelani5222
9 ай бұрын
Thank you for existing MITOCW and Prof. Gilbert Strang.
@uzferry5524
Жыл бұрын
bruh the fibonacci example just blew my mind. crazy how linear algebra just works like that!!
@Huayuan-p4z
9 ай бұрын
I have learned about the Fibonacci sequence in my high school, and it is so good to have a new perspective on the magical sequence.I think the significane of learning lies in the collection of new perspectives.😀
@Zumerjud
9 жыл бұрын
This is so beautiful!
@dwijdixit7810
Жыл бұрын
33:40 Correction: Eigenvalue matrix be multiplied to S from the right. That has been made in the book. Probably, it slipped off Prof. Strang in the flow.
@shadownik2327
8 ай бұрын
Now I get it, so its like breaking the thing ( vector or matrix or system really) we want to transform into little parts and then transforming them individually cz thats easier as the parts get transformed in the same direction and then adding up all those pieces. E vectors tell us how to make the pieces and e values how to make the transformation with the given matrix or system. Wow thanks ! It’s like something fit in in my mind and became very simple. Basically this is like finding the easiest way to transform. Thanks to @MIT and Professor Strang for making this available online for free.
@zyctc000
9 ай бұрын
If any one ever asks you about why the Fibonacci and the golden ratio phi is connected , point him/her to this video. Thank you Dr. Strang
@ccamii__
Жыл бұрын
Absolutely amazing! This lecture really helped me to understand better the ideas about Linear Algebra I've already had.
@starriet
2 жыл бұрын
Notes for future ref.) (7:16) there are _some_ matrices that do _NOT_ have n-independent eigenvectors, but _most_ of the matrices we deal with do have n-independent eigenvectors. (17:14) If all evalues are different, there _must_ be n-indep evectors. But if there are same evalues, it's possible _no_ n-indep evectors. (Identity matrix is an example of having the same evalues but still having n-indep evectors) * Also, the position of Lambda and S should be changed(32:36). You'll see why by just thinking matrix multiplication, and it can also be viewed by knowing A^100=S*Lambda^100*S^-1 and u_0=S*c. Thus, it should be S*Lambda^100*c, and this also can be thought of as 'transformation' between the two different bases - one of the two is the set of the egenvectors of A. * Also, (43:34) How prof. Strang could calculate that?? Actually that number _1.618033988749894..._ is called the 'golden ratio'. * (8:15) Note that A and Lambda are 'similar'. (And, S and S_-1(S inverse) transforms the coordinates.. you know what I mean.. both A and Lambda can be though of as some "transformation" based on different basis.. and S(or S_-1) transforms the coord between those two world.)
@deveshvarma8531
2 жыл бұрын
I spent a few hours on the second point before figuring it out :(
@aattoommmmable
13 жыл бұрын
the lecture and the teacher of my life!
@serden8804
4 жыл бұрын
bro yasiyor musun
@bastudil94
10 жыл бұрын
There is a MISTAKE on the formula of the minute 32:31. It must be S(Λ^100)c in order to work as it is supposed. However it is an excellent lecture, thanks a lot. :)
@YaguangLi
10 жыл бұрын
Yes, I am also confused by this mistake.
@sammao8478
9 жыл бұрын
Yaguang Li agree with you.
@AdrianVrabie
8 жыл бұрын
+Bryan Astudillo Carpio why not S(Λ^100)S^{-1}c ???
@apocalypse2004
8 жыл бұрын
u0 is Sc, so S inverse cancels out with the S
@daiz9109
7 жыл бұрын
You're right... it confused me too...
@eren96lmn
8 жыл бұрын
43:36 that moment when your professor's computational abilities goes far beyond standart human capabilities
@BalerionFyre
8 жыл бұрын
Yeah wtf? How did he do that in his head?? lol
@BalerionFyre
8 жыл бұрын
Wait a minute! He didn't do anything special. 1.618... is the golden ratio! He just knew the first 4 digits. Damn that's a little anticlimactic. Bummer.
@AdrianVrabie
8 жыл бұрын
+Stephen Lovejoy Damn! :D Wow! AWESOME! I have no words! Nice spot! I actually checked it in Octave and I was amazed the prof could do it in his head. But I guess he knew the Fibonacci is related to the golden ratio.
@IJOHN84
5 жыл бұрын
All students should know the solution to that golden quadratic by heart.
@ozzyfromspace
4 жыл бұрын
Fun fact since we're all talking about the golden ratio. The Fibonacci sequence isn't that special. Any sequence F_(k+2) = F_(k+1) + F_k for any seeds F_0 = a and F_1 = b != -a generate a sequence that grows at the rate (1+sqrt(5))/2 .. your golden ratio. Another fun way to check this: take the limit of the ratio of numbers in your arbitrary sequence with your preferred software :) edit: that's a great excuse to write a bit of code lol
@benzhang7261
4 жыл бұрын
Master Yoda passed on what he has learnt by fibonacci and 1.618.
@maoqiutong
Жыл бұрын
32:41 There is a slight error here. The result Λ^100 * S * C may be wrong. I think it should be S * Λ^100 * C.
@wendywang4232
12 жыл бұрын
something wrong with this lecture, 32:39, A^{100}u_0=SM^100c. Here I use M to substitute the eigenvalue diagonal matrix. The professor said A^{100}u_0=M^100Sc which is not correct.
@ozzyfromspace
4 жыл бұрын
Did we ever prove that if the set of eigenvalues are distinct, the set of eigenvectors are linearly independent? I ask because at ~ 32:00 taking u_o = c1*x1 + c2*x2 + ... + cn*xn requires the eigenvectors to form a basis for an n-dimensional vector space (i.e. span the column space of an invertible matrix). It feels right but I have no solid background for how to think about it
@roshinis9986
10 ай бұрын
The idea is easy for 2d. If you have two distinct eigenvalues and their corresponding eigenvectors, you don't just have one eigenvector per eigenvalue, the whole span of that vector (its multiples forming a line) are also the eigenvectors associated with that eigenvalue. If the original eigenvectors were to be dependent, they would lie in the same line making it impossible for them to scale by a factor of two distinct eigenvalues simultaneously. I haven't yet been able to extend this intuition to 3 or higher dimensions though as now dependence need not mean lying in the same line.
@jeanpierre-st7rl
6 ай бұрын
@@roshinis9986 Hi @ 29:46 Uo = C1X1 + C2X2 +C3X3... Is U0 a vector? If so, How can split this U0 in to a combination of eigen vectors? What is Ci ? If you have any info pleases let me know. Thanks.
@muyuanliu3175
20 күн бұрын
32:42, should be S lambda^100 c, great lecture, 3rd time I learn this
@alexspiers6229
6 ай бұрын
This is one of the best in the series
@florianwicher
6 жыл бұрын
Really happy this is online! Thank you Professor :)
@Afnimation
11 жыл бұрын
well i got impressed at the begining, but when he stated the second eigenvalue i realized it is just the golden ratio... That does not demerits him, he's great!
@cecilimiao
14 жыл бұрын
@cuinuc I think they are actually the same, because LAMBDA is a diagonal matrix, you can have a try.
@RolfBazuin
11 жыл бұрын
Who would have guessed, when this guy explains it, it almost sounds easy! You, dear dr. Strang, are a master at what you do...
@dexterod
8 жыл бұрын
I'd say if you play this video at speed 1.5, it's even more awesome!
@abdulbasithashraf5480
5 жыл бұрын
Trueee
@ozzyfromspace
4 жыл бұрын
1x all the way. I savor the learning ☺️
@Mohamed1992able
13 жыл бұрын
a big thanks tothis prof for his efforts to give us cours about linear algebra
@gomasaanjanna2897
3 жыл бұрын
Iam from india I love your teaching
@abdulghanialmasri5550
2 жыл бұрын
The best math teacher ever.
@kunleolutomilayo4018
5 жыл бұрын
Thank you, Prof. Thank you, MIT.
@mospehraict
13 жыл бұрын
@PhilOrzechowski he does it to make first order difference equations system out of second order
@PaulHobbs23
13 жыл бұрын
@lolololort 1/2(1 + sqrt(5)) is also the golden ratio! Math is amazing =] I'm sure the professor knew the answer and didn't calculate it in his head on the spot.
@chiaochao9550
3 жыл бұрын
46:08 It should be F_100 is similar to c_1 * lambda1 * x_1. The professor missed x_1 here. But if you assume x_1 is 1 (which is the case here), then this is correct.
@starriet
2 жыл бұрын
Not actually. F_100 is a number and x_1 is a vector.
@jasonhe6947
4 жыл бұрын
absolutely a brilliant example for how to apply eigenvalues to real world problem
@gianlucacococcia2384
4 жыл бұрын
Can I ask you why A1 x x1 is lambda x1?
@gianlucacococcia2384
4 жыл бұрын
Zhixun He
@sharmabu
Ай бұрын
absolutely beautiful
@eugenek951
7 ай бұрын
He is my linear algebra super hero!🙂
@technoshrink
9 жыл бұрын
U0 == "you know it" First time I've heard his boston accent c:
@iDiAnZhu
11 жыл бұрын
At around 32:45, Prof. Strang writes Lambda^100*S*c. Notation wise, shouldn't this be S*Lambda^100*c?
@jojowasamanwho
Жыл бұрын
19:21 I would sure like to see the proof that if there are no repeated eigenvalues, then there are certain to be n linearly independent eigenvectors
@hektor6766
5 жыл бұрын
You can hear the students chuckling as they recognized the Golden Ratio. Didn't quite recognize it as (1 + root 5)/2.
@rambohrynyk8897
10 ай бұрын
It always shits me how quickly the students clammer to get out of the class….how are you not absolutely dumbfounded by the profundity of what this great man is laying down!!!!
@pelemanov
13 жыл бұрын
@LAnonHubbard You don't really need to know about differential equations to understand this lecture. Just watch lessons 1 to 20 as well ;-). Takes you only 15h :-D.
@iebalazs
2 жыл бұрын
At 32:32 the expression is actually S*Lamdba^100*c, and not Lambda^100*S*c .
@praduk
15 жыл бұрын
Fibonacci numbers being solved for as an algebraic equation with linear algebra was pretty cool.
@Zoro3120
8 жыл бұрын
In the computation of the Eigen values for A², he used A = SʌSˉ¹ to derive that ʌ² represents its Eigen value matrix. However this can be true only if S is invertible for A², which need not be always true. For example, for the matrix below (say A), the Eigen values are 1, -1(refer previous lecture). This would imply that A² has only one Eigen value of 1. This would imply that S has 2 columns which are same (if it has only one column then it is no longer square and hence inverse doesn't apply) and hence non invertible. This implies that this proof cannot be used for all the cases of the matrix A. _ _ │ 0 1 │ │ 1 0 │ ¯ ¯ Is there something I'm missing here?
@hinmatth
7 жыл бұрын
Please check 17:32
@LAnonHubbard
13 жыл бұрын
I've only just learnt about eigenvalues and eigenvectors from KhanAcademy and Strang's Lecture 21 so a lot of this went whoooosh over my head, but managed to find the first 20 minutes useful. Hope to come back to this when I've looked at differential equations (which AFAIK are very daunting), etc and understand more of it.
@rolandheinze7182
5 жыл бұрын
Don't think you need diff EQ at all to understand the algebra. Maybe the applications
@joe130l
3 жыл бұрын
so it seems like the professor emphasized the importance of the eigenvalue here, that's nice. but is the eigenvector of any importance? what's a good example of eigenvectors?
@sammatthew7
16 күн бұрын
GOLD
@richarddow8967
Жыл бұрын
beautifully simple how that Fibonacci worked out
@dennisyangji
14 жыл бұрын
A great lecture showing us the wonderful secret behind linear algebra
@lastchance8142
2 жыл бұрын
Clearly Prof.Strang is a master, and his lectures are brilliant. But how do the students learn without Q&A? Is this standard procedure at MIT?
@jamesmcpherson3924
4 жыл бұрын
I had to pause to figure out how he got the eigenvectors at the end. Plugging in Phi works but it wasn’t until I watched again that I noticed he was pointing to the lambda^2-lambda-1=0 relationship to reveal the vector.
@mdrumon319
4 ай бұрын
I didn't get , how Au =ΛSc is right. Shouldn't Λ be at right of S. Is there anyone who can help ?
@thomassun3046
3 ай бұрын
Here comes a question, How U0 is equal to c1x1+c2x2...+cnxn. at 29:50, confused, could anyone explain it to me?
@mike-yj5mm
3 жыл бұрын
I don't understand 11:25 why A square can be written in the way on the blackboard. I think A^2 should be (S Lambda S^-1)^T (S Lambda S^-1), the result differs from the one on the blackboard. Could someone explain this?
@mike-yj5mm
3 жыл бұрын
Okay, I figured it out. The S is an orthogonal matrix under the n independent eigenvector assumption, the inverse of which equals to its transpose.
@APaleDot
Жыл бұрын
@@mike-yj5mm No, it doesn't require S to be an orthogonal matrix. n independent eigenvectors ≠ n orthogonal eigenvectors of unit length, which would be required to make S an orthogonal matrix. At this point in the lecture we've already proven that A = S ∧ S^-1 and therefore it follows immediately that A^2 = AA = (S ∧ S^-1)(S ∧ S^-1). All the matrices are square, so there is no conflict in their dimensions.
@tomodren
12 жыл бұрын
Thank you for posting this. These videos will allow me to pass my class!
@MeridianLights
8 жыл бұрын
In the Fib example, it seems impossible to find a c1 and c2 s.t. c1*x1 + c2*x2 = [1 0]
@jasarinvorawathanabuncha6620
8 жыл бұрын
You can! Try using elimination to easily see that c1 = 1/sqrt5 c2 = -1/sqrt5
@ozzyfromspace
4 жыл бұрын
@@jasarinvorawathanabuncha6620 not true, c1 = (a-1)*(1+(a-1)/(b-a)) and c2 = -(a-1)*(b-1)/(b-a) where a is the positive eigenvector and b is the negative eigenvector of our problem. Also worth noting, the eigenvectors have the form x = [1/(lambda - 1), 1], not x = [-lambda, 1] as the professor wrote :) There were a few mistakes in the way to the solution so whatever answer we arrived at was simply not correct Lol I just watched the video now but obviously this is really late for you, hopefully someone else finds this useful. Best wishes friend
@tanjiaqing1294
4 жыл бұрын
@@ozzyfromspace [1/(lambda - 1), 1] = [-lambda, 1] in this case, u can plug in lambda = (1+sqrt(5))/2 and (1-sqrt(5))/2 to check. So the c1 and c2 got by @Jassarin should be correct.
@zhiweithean800
9 ай бұрын
But what is c1 and c2 at the end? c1x1 + c2x2 = [1 0]. I cant find c1 and c2 with x1 and x2 that will produce [1 0]. Am I missing something? What is the value of c1 and c2?
@amyzeng7130
2 жыл бұрын
What a brilliant lecture !!!
@kebabsallad
13 жыл бұрын
@PhilOrzechowski , he says that he just adds it to create a system of equations.
@NisargJain
6 жыл бұрын
Never in my entire life, i would have been able to convert that Fibonacci sequence in matrix form. Untill, he did it.
@thedailyepochs338
4 жыл бұрын
can you explain how he did it
@NisargJain
4 жыл бұрын
@@thedailyepochs338 sure, for understanding that we must first understand what fibonacci sequence is, in a fibonacci sequence every term is the sum of previous two terms (given the first two terms starting from 0 and then 1). So, the 3rd term F3= F2+F1=1+0=1. Similarly F(k+2)= F(k+1) + F(k). But in a matrix, let us assume that U(k) is 2 dimensional vector that consists of the first term as F(k+1) and 2nd term as F(k). Similarly U(k+1) would be [F(k+2), F(k+1)]. But see that the first term of U(k+1) that is F(k+2) is equal to sum of the term of U(k) that F(k+1) + F(k); and the second term of U(k+1) is is F(k+1) which is only the first term of U(k). Hence we get the matrix A=[ 1 1 ] [ 1 0] If you multiply, AU(k) which is A U(k) [ 1 1 ] [ F(k+1) ] [ 1 0] [ F(k) ] You will fet first term as sum of the terms of U(k) and second term as just the first term of U(k) which we deduced to be U(k+1).
@thedailyepochs338
4 жыл бұрын
@@NisargJain thanks , really appreciate it
@phononify
Жыл бұрын
very nice discussion about Fibonacci ... great !
@tchappyha4034
5 жыл бұрын
His lecture is very good. But his lecture is not complete. He didn't prove the fact that n eigenvectors are linearly independent if the matrix A has n different eigenvalues. He lectured only the case in which the eigenvectors are linearly independent. He always omits difficult cases.
@thedarkkhan
8 жыл бұрын
-0.618 1 1 -1.618 The Null Space of the given matrix should be the zero vector because the RREF will become: 1 0 0 1 Which means the Null Space(and hence the Eigenvector) is the zero vector for the first Eigenvalue? Correct me if I'm wrong?
@chaitanyapatel1946
7 жыл бұрын
RREF is wrong. It's not identity. It has zero bottom row, hence rank 1. Dimension of null space is 1 and hence one non zero vector for first eigen value.
@anonymous.youtuber
3 жыл бұрын
Just wondering...what keeps us from calling the eigenvector matrix E instead of S ? Is E already used for something else ?
@Itsmeaboud
3 жыл бұрын
Yes, it is used for elimination matrix.
@ashutoshtiwari4398
5 жыл бұрын
Why the skew-symmetric matrix have zero or imaginary eigenvalue?
@sidaliu8989
5 жыл бұрын
By the definition of the skew-symmetric matrix (A^T=-A), all entries in the diagonal of the matrix must be 0. So when we come up with the characteristic equation, it will be lambda^n+b^2=0 (since the trace is zero and the determinant is some square), and this will give us pure imaginary solutions if b^2>0.
@codeo6246
2 жыл бұрын
Can't you also use the determinant to figure out that Aᵏ → 0 as K → ∞ ? i.e. if det A < 1 then Aᵏ → 0
@АлександрСницаренко-р4д
3 жыл бұрын
MIT, thanks you!
@abhi220
7 жыл бұрын
I have a doubt in Difference equations part. He writes u_0 as a combination of eigen-vectors of A. Why should this be true?
@olfchandan
7 жыл бұрын
eigen vectors span the entire space (Remeber - S is square invertible matrix). So, U0 will be a linear combination of eigen vectors.
@suziiemusic
7 жыл бұрын
A set of n independent eigenvectors, each one with n components, is a basis for Rn, and therefore any vector in Rn (including u0) can be written as a linear combination of these n eigenvectors. We could choose any other set of n independent vectors as a basis and do the same thing. The "standard" basis would be the columns of the identity matrix, which in 3 dimensions correspond to the x,y and z axes.
@shamsularefinsajib7778
11 жыл бұрын
Gilbert strang a great math teacher............
@valtih1978
10 жыл бұрын
In the concluding "review" he reminds us that initial state vector u₀ must be decomposed into eigenvectors. However, he seems refused to say that the eigenvector will look like exponential function [1 λ λ² λ³ ] of eigenvalues λ, valjok.blogspot.com/2013/12/eigenvectors-of-recurrence-equations.html. In case you start evolving from y₀ !=1, instead of [y₀ y₁ y₂ y₃ …] = [1 λ λ² λ³], your recursive process evolves through the state variables [y₀ y₀λ y₀λ² y₀λ³ ]. Yet, y₀ is unimportant as long as eigenvector is concerned. Only eigenvalues λ are important. It is cool that [1 λ λ² λ³] is the exponential vector and since exponents must be the solutions of linear differential equations. I just do not understand how why does he explain how our recursion evolves u₀ → u₁ → … (the initial value problem) if our goal is to find all solutions, regardless of what are the boundary conditions.
@Hindusandaczech
13 жыл бұрын
Bravo!!! Very much the best and premium stuff.
@stumbling
8 жыл бұрын
7:33 Surprise horn?
@mayurkulkarni755
7 жыл бұрын
wtf was that xD
@canmumcu9804
5 жыл бұрын
probably grinding of a chair xd
@puneetgarg4731
Жыл бұрын
I am a lil confused what is the value of A^2? is it A.A or is it a(A transpose )A
@khanhdovanit
3 жыл бұрын
15:02 interested information inside matrix - eigenvalues
@suzukikenta1079
8 жыл бұрын
Could someone explain why vector, lambda and 1 is a solution to the nullspace at 48:43? First component, (1-lambda)*lambda+1 is not 0 or different from lambda^2 - lambda -1.
@antoniolewis1016
8 жыл бұрын
Not sure what you mean, but the professor was saying the vector [lambda ] [ 1 ] is in the nullspace of the matrix A-lambda* I, and not the nullspace of A. And since this vector is in that nullspace, it must be an eigenvector of A attached to eigenvalue lamda.
@jasarinvorawathanabuncha6620
8 жыл бұрын
(1-lambda)*lambda+1 is indeed 0 because of the determinant next to it: lambda^2 - lambda -1= 0
@roronoa_d_law1075
6 жыл бұрын
that makes lambda^2 - lambda + 1 and not -1 I'm confused
@roronoa_d_law1075
6 жыл бұрын
nevermind
@rolandheinze7182
5 жыл бұрын
Do the matrix multiplication and see fr yourself, it evaluates to 0 fr the eigenvalues. Also these eigenvectors give you the same equation as characteristic equation (flipped to other side of 0=...) So must be a solution
@healthfreak5438
9 жыл бұрын
why are we representing mu(k) vector as the combination of eigen vectors in the problem of Fibonacci sequence?
@utxeee
6 жыл бұрын
So, the second component of u(k+1) is useless, right? The actual value is given by the first component.
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