Diagonalization
ฝัง
- เผยแพร่เมื่อ 9 ก.ค. 2024
- Now that we know about eigenvalues and eigenvectors, we are ready to learn about diagonalization. This involves writing a matrix as a product of other matrices, one of which is a diagonal matrix, with values only along the main diagonal. This has many applications in computation, so let's take a look!
Script by Howard Whittle
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This is always a hard topic to teach. This is straight forward and clear. Great video!
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This definitely simplified everything that was taught to me in my class lecture. Everything is super clear now. Thank you so much!
Prof Dave: I am currently doing self-study of every math course required for an undergraduate math program and I was having a hell of a time understanding fully how to perform diagonalization! I have read countless textbooks' sections on diagonalization and watched several other videos. I took thorough notes from your mini lecture, followed along with you on the example and am stoked to say I was able to go through the example problem at the end and got everything right. ALSO...nowhere else has anyone mentioned to ALWAYS choose x2=1. That tiny detail helped make everything else click and I agree with you that the process of diagonalization is in fact easy, albeit time-consuming. I cannot thank you enough for this! The way you go through and show every single detail is a TREMENDOUS help! I really appreciate you!
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How is it that I sat through two lectures on diagonalisation and it hardly made sense, yet after watching this video the entire concept could NOT be any simpler??
Actual lifesaver, I'm defs getting 100% on my linear algebra quiz tomorrow
7:41 "so, lets check comprehension" wowww, never seen that part in any other video. awesome, thanks a lot for including that :)
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Thank you, Prof Dave. This video made me clear in this topic otherwise some of text books written in Japanese are so hard to understand for newbie.
5:19 How will we know that in which sequence we put the eigen vectors ,is there any rule of it or we can just randomly put any eigen vector in first coloum or second column.?
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I reading my linear algebra book but i cant comprehend .Looking for several videos but im still confuse halfway..
This is the video that enable me to understand it clearly.
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Thank you so much professor for your explanation. Good luck
Great video! This helped me so much. I know this is a little late but I just wanted to point out that instead of calculating X inverse to check your answer at the end, you can simply check to see if AX = XD . This is much easier if you are asked a similar question that uses nxn matrices where n > 2 as the computing the inverse becomes more annoying as n increases.
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It really helps me! Thanks a lot!
4:13 , why did you choose x2 = 1, why not x1 = 1?
If we take x1 = 1 will our answer be the same ?
Great explanation! Thanks!
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@4:48 Why can we just choose x2 = 1? Don't you get x2 by subtracting 5x and dividing 4 giving you x2= 5x/4 ?
very clear explanation!
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How do you know which eigen vector to use first and second to make up the eigen vector matrix?
How did you just make me understand this so easily :D
Prof when i try the example in the end the inverse reads -1,-2,-2 and -2 since we multiplied X by -2...I have been stuck on this for hours..can you please explain it?
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Great explanation
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what do you do if thie L is only equal to one number like for example "L^2+4L+4" will be only equal to -2. Would I use the same one to make two of the same thing? like I get [3,1] for one of my eigen vectors, then would I just write the same thing for my 2nd eigen vector [3,1]
When I multiplied X^-1DX, the result was not A it was A with the -1 and 2 switched places. Did I do something wrong or is this how it is supposed to be?
It was X(D)X^-1 as matrix multiplication is not commutative
Where was this series the beginning of the semester? I now have to cram so much information the days before my final.
great explaination
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Thanks so much prof
Hi Professor Dave,
Thanks for the great linear algebra videos.
At around 7:00, why did you multiply D by X inverse first? Instead of multiplying matrix X by matrix D first? I didn't think that matrix multiplication was commutative?
not being commutative simply means that we can't reverse the order in which we list the matrices, but we can simplify in any order we want, if you do the other multiplication first you should get the same answer
excellent explanation
Hi, do you have a video to explain why this works?
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Please why do you normally choose X2=1
I don’t get that part
Thanks prof. Dave for your amazing comprehensive but precise talk on linear algebra. In this particular video why are we supposed to take X2 equal to 1 for every time as you were taking? Is this a free variable that allows us to assign any value to it or just for ease calculation as well! Thanks in advance.
did u get this ans? i wanna know too
@@tinyasira6132 its because it is a free variable, if you reduce to echelon form you'll see that the only pivot will be x1, therefore the x2 will be a free variable.
Thank You!🥰🥰
Great explanation sir
At the last one I had the same eigenvalues 4 and 5 but I turned them around in matrix D. Is that wrong is doesnt it matter?
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Professor Dave casually saving everyone's ass again this year
professor dave, how do you know how to put the 1 and 2 in that order in the diagonal matrix?
wish he answered this
Super video!
sir in integration video you say that we can find the area of a curve by slipting into a bunch of rectangles a mountain has a curve shape, can we split it in to a bunch of rectangles, please reply
yes, the rectangles are infinitely thin so they are essentially lines, but it would be nearly impossible to find a function that corresponds with the topology of a mountain so i don't think it's really meant for that application
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where do the -4's go at 3:57?
Is there only two videos relating to the eigen topic?
Really talented.keep it up.
thanks! but i keep finding X inverse's last row as [ -1,-1 ] instead of [ 5,5 ] at 6:26. I am using the method of reducing the A with its Identity matrix. I also used some online matrix inverse calculators, also they are giving me my result instead of this video's. Are they just mean same thing? Or the video is not correct?
Can't thank you enough.
Sir, If X2 corresponds to lamda=1 and X1 corresponds to lamda=2 do we get same diagonal matrix?
Can you explain what XDX-1 means? I understand XD is a matrix of the scaled eigenvectors after the transformation A and X-1 is the inverse of X. However, I don't know what it means for a matrix to be the inverse of the eigenvectors.
If you want to fully understand it, I suggest you watch 3blue1brown's videos about linear algebra, but here is the short version. The matrix A is a matrix that transforms space, everything gets shifted except for the eigenvectors, they just get longer. So what we can do is first map the eigenvectors to our unit vectors (X^-1), then scale those vectors (D), then map them back to the eigenvectors (X). Since these are all linear transformations, this is not only true for the eigenvectors, but also for all other vectors in the 2D plane. So XDX^-1 = A.
Perfect
P.S. in 6:45 ( A= X-1 D X ) is known as eigenvalue decomposition
Great Professor
what if for 3 by 3 matrices I got 2 lemda value then how to solve that problem?
amazing I understand BST ❤ thnks
why do they choose the value of x2 = 1 for that equation?
Sir can I write eigen vectors at any place during making X matrix
How pure complex eigenvalues of rotation matrix cause rotation by actual elongation with factor lets say 'i' ? Should i think complex plain as 4 dimensional to grasp it intuitively because it doesnt make sense at all like how multiplying complex eigenvector by unit 'i' will scale it in a way so its projection on real plane will be seen as rotation of components of real vectors?
excellent sir
God Bless you Sir
THANK YOU
can my D X be in a different order as long as their eigen value/vectors are in the same order respectively to each other?
Yes.
thnak you so much
Thank u professor
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I have question does D has to be in order of
1 0
0 2
or it doesn't matter if it was
0 2
1 0 ?
The values in D are the eigenvalues, order them from lowest on the top left to highest on the bottom right, following the diagonal :) (As shown from 7:58 and onwards)
@@Entervation thnx a lot