as a native french speaker, I understand your videos better than most of any french courses I've read / watched ! Thanks a lot, you save me a lot of time and desperation :D
hate it. Seems more theory rather than intuition. Let alone super rigorous dumbed down overrated book such as axler's. All students would end up doing would be memorizing them for the sake of getting through all the complexity since the intuition is nowhere to be found
I've been learning eigen values and vectors solved a bunch of problems without even understanding what i was doing....Thanks a lot for that explanation!!!
last 30 secs taught me more than last 3 months. Thanks you sir. Your way of teaching is impeccable. I am absolutely stunned by the last minute intution.MIND = BLOWN
I have searched and searched for an explanation like this one, took me months to finally found an explanation that anyone can understand. You are a talented teacher, thank you!
You have not just made my day but my career. I have following you for two days and seems u just keep cracking the rocket science. Am doing an Msc Data Science. Thank u so much
How this man has not blown up bigger that someone like blackpenredpen is beyond me. I am in Calculus II right now, and this video made perfect sense to me.
Just came across your channel .. Your videos are absolutely amazing! I'm in a multivariate analysis course, where I need to refresh my linear algebra skills, so these videos are really helpful.
Your presentation skills are top-notch. Since this is the first of your videos I've watched, I don't yet know whether you devote another video to other properties of eigenvectors. You stress the collinearity, but don't talk about the way the hypervolume of some set of vectors collapses. Maybe you do this in a video where you define the determinant. Maybe your mentioning the null space of the matrix covers this. At any rate, I'll say at this point that I'll probably find all your presentations worthwhile. Best wishes in growing your channel.
These are awesome videos! They really Intuitively connect theoretical concepts in linear algebra with application in ways that I was never explicitly taught! Keep up the great work!
I haven’t had a reason to dive into this kind of topic for over 20 years: only saw it during undergrad & grad school. But I enjoyed your technique of going through it.
Imagine squishing or stretching a balloon by pressing on it. Certain parts of the balloon will always expand in specific directions, even if the whole shape changes-that’s like how eigenvectors and eigenvalues show up when transformations are applied. So, eigenvectors give us those "directions that don’t change direction," and eigenvalues tell us how much things get stretched or shrunk along those directions! Simple Example -- Suppose we have a dataset with information on height and weight for a group of people. By calculating the eigenvectors and eigenvalues of this dataset, we can: Find the direction in which height and weight are most correlated (an eigenvector showing the pattern). See how important this correlation is with an eigenvalue, which tells us if this pattern captures a big or small portion of the data's overall "shape."
I think it's important to point out what an operator can do to a vector (A*x) in general, and then point out that these eigen directions are special, because here the operator's effect is just scaling. And this is useful, because...
Take the last section of the video, knowing that eigen is a German word for "own" and you will never forget what is the importance of eigenvalues and eigenvectors.
Good explanation of the math. But for 40 years I still struggle with what eigenvalues really are. Your fish example was better than most I have heard but I am still missing something vital.
It's definitely a tricky concept and I'm glad this video helped a little bit. Took me a long time to understand too. I think the easiest explanation is that an eigenvector is one where the matrix will map a vector to a multiple of itself (so that the input vector and the output vector both point in the same direction). Why does this matter? Because the same direction ensures the same ratios between each individual vector component which loosely means that the input and output vectors have the same proportions.
@@ritvikmath That helps. I have come across it in PCA and in quantum mechanics. (Yes, I am eclectic). Another question: do eigenvalues HAVE to be real?
How does this property of a vector (eigen vector) remains in the same dimension even after transformation (by A) helps in some problem solving (related to ML)?
All teachers seem to fail at the stage where they include the identity matrix [02:07] why is that? The reason is, because they understand that putting the identity matrix in does not affect vector or lambda. But they never tell you this vital bit of information. And they still wonder why people fail to understand mathematics. They had to learn it themselves. but they are not including this vital piece of knowledge in their explanations. It now seems a trivial point to them, but for a student starting out, it is not trivial. In fact, for any student with a basic understanding of Algebra, they would wonder why I only one side of the equation Ax = λx being multiplied by I, the identity matrix, surely this breaks the rule of algebra which says whatever you multiply one side of the equation by, you must multiply the other side of the equation by. And yet here, we see only one side of the equation being multiplied by I, the identity matrix. Without any explanation as to why you can do that. It's time you start explaining why it is alright to multiply λx by I, the identity matrix on one side and not the other side of the equation: answer, because as any identity does, it does not change the number. hence the word identity. [02:23] let's subtract, Ax - λx = λIx, ready? Ax - λx -[λIx]= λIx - [λIx], okay how does that equal Ax - λIx = 0? well, λIx - [ λIx] equals zero, so the right hand side of the equation is fine. but what about the left hand side? What we have is Ax - λx -[λIx] . Okay so let's apply a little algebra: like terms can be added or subtracted. No like terms so, nothing can be subtracted here. Amazingly, ritvikmath seems to think these can be subtracted. Actually, in his calculations the term λx just magically disappers, so he is left with Ax - λIx = 0. He could have got to this result a different way. Let's start out with Ax = λx, then subtract λx from both sides (as laws of algebra suggest) that would give: Ax - λx = λx - λx. which results in, Ax - λx = 0. Now he has a choice to include the identity matrix Ax - λIx = 0. see, same result. Nothing magical, nothing disappears, every step accounted for. His main argument is right. And I look forward to his video on determinants of matrices, i.e. proving that a matrix is non-invertable.
funny how the mathematical understanding behind it is very important to grasp, however we will never have to calculate the eigenvectors and values by hand after university.
I have a quick question, at 1:01 you mention that lambda is such that it is a real number, can't this be extended to imaginary numbers as well? Btw, Thanks for you great work!
In aerodrums screens eigen vector and eigenvalues for different landing planes may be manipulated with out collision by having graphics accordingly correct? That might have been a better explanation.
Math and engineering classes always seem to treat Ax = λx as an abstraction. I wish someone would say at the beginning of the discussion that Ax = λx means that an eigenvector is a vector that points in the same direction after it's been operated on by A.
@@s25412 My comment about direction was a generality. Of course, A might transform x so that it points in the opposite direction, but the eigenvector will point along the same line as it was pointing before being operated on by A. A scalar multiple of an eigenvector is also an eigenvector.
hey, can we subtract the mean of each column from the column so as to make it zero mean before calculating the cov matrix. and in some textbooks it is divided by n-1 instead of n. why is that? Thanks
It is because of the difference between "population" and "sample" if you use for population then the accuracy must be considered so that we use n-1 it's for more accuracy.
as a native french speaker, I understand your videos better than most of any french courses I've read / watched ! Thanks a lot, you save me a lot of time and desperation :D
Je suis heureux d'aider! (Sorry I only took 3 years of french in high school 😁)
Last 2-3 mins are invaluable, I knew eigen values and eigenvectors but didnt know where to use them. Thanks very much!
Same here, cheers to the author of the video!
You have an easy, inclusive and coherent way of teaching. Great job!!👌
Thank you! 😃
And the best explanation eigenvector/value prize goes to.... this guy!... good job man ...great video
You are amazing ... anyone who is watching this video please don't miss last few minutes.
*me learning linear algebra for the first time even though i passed the class three or four years ago*
Oof
lol
hate it. Seems more theory rather than intuition. Let alone super rigorous dumbed down overrated book such as axler's. All students would end up doing would be memorizing them for the sake of getting through all the complexity since the intuition is nowhere to be found
I've been learning eigen values and vectors solved a bunch of problems without even understanding what i was doing....Thanks a lot for that explanation!!!
last 30 secs taught me more than last 3 months. Thanks you sir. Your way of teaching is impeccable. I am absolutely stunned by the last minute intution.MIND = BLOWN
Great to hear!
I have searched and searched for an explanation like this one, took me months to finally found an explanation that anyone can understand. You are a talented teacher, thank you!
there's a really old maths book by Bostock and Chandler that has a good explanation of this too.
You have not just made my day but my career. I have following you for two days and seems u just keep cracking the rocket science. Am doing an Msc Data Science. Thank u so much
How this man has not blown up bigger that someone like blackpenredpen is beyond me. I am in Calculus II right now, and this video made perfect sense to me.
That's the goal :)
Just came across your channel .. Your videos are absolutely amazing! I'm in a multivariate analysis course, where I need to refresh my linear algebra skills, so these videos are really helpful.
No problem!
which analysis course you doing?
Your presentation skills are top-notch. Since this is the first of your videos I've watched, I don't yet know whether you devote another video to other properties of eigenvectors. You stress the collinearity, but don't talk about the way the hypervolume of some set of vectors collapses. Maybe you do this in a video where you define the determinant. Maybe your mentioning the null space of the matrix covers this. At any rate, I'll say at this point that I'll probably find all your presentations worthwhile. Best wishes in growing your channel.
Thank you! I took linear over a year ago and your explanations clear up so many questions I had.
Great to hear!
You not only explained the math operations, but thanks for the insite of why we are doingit. Thanks for the enlightment.
No problem!
Many thanks. You summarised important chapter of linear Algebra in just less than 12 minutes.
This is way better than the explanation I had in my linear algebra course long ago!
It would be useful to be able to like these videos more than once to express how appreciated they are for a newbie! Thank you!
Really did a good job man. Appreciate your time and valuable information
11:35 useful to think about the same "ratio", thank you boss
this just made me so happy... THANK U!
Just came to your vids by accident.. now I'm asking how I could not have got here earlier..!
Excellent explanation. Very useful in my mathematical modelling of infectious disease learning. Thank you
This man is a real gem.
I love your videos .. super helpful not just to refresh the knowledge but also understand it in a more intuitive way!! Thank you so much !
No problem!
These are awesome videos! They really Intuitively connect theoretical concepts in linear algebra with application in ways that I was never explicitly taught! Keep up the great work!
I haven’t had a reason to dive into this kind of topic for over 20 years: only saw it during undergrad & grad school. But I enjoyed your technique of going through it.
Thanks!
Wow, I don’t know why professors rarely provide motivation like this.
this cleared up so much and importance of why we need eigenvectors, tysm!!
Very clean and clear presentation.
You're helping me a lot refreshing these concepts! So happy I found your channel!
Awesome! Thank you!
You are a literally a Godsend and a savior, Machine Learning is becoming more clear with every video i watch of you fam, thank you!
You are good. I knew some linear algebra but I couldn't get the "feel" of it. Watching this Data Science Basics series has changed it.
Your videos are absolutely top-notch. Keep it up!
Best video on eigen
great explanation. thanks.
Awesome splendid mesmerising
this video made it all look so so simple, thankyou very much!
thank you, I learned faster and easier with your explanation, you rock!
Which video to follow for the importance of invertibility?
I appreciate you making these videos as they have helped so much in understanding abstract machine learning/data science concepts! :) Cheers to you!
Cheers to me . I taught him
What an amazing example in eigenvector. Help fish to find fish with same "figures". Surely has been used in dating apps😂
Imagine squishing or stretching a balloon by pressing on it. Certain parts of the balloon will always expand in specific directions, even if the whole shape changes-that’s like how eigenvectors and eigenvalues show up when transformations are applied.
So, eigenvectors give us those "directions that don’t change direction," and eigenvalues tell us how much things get stretched or shrunk along those directions!
Simple Example --
Suppose we have a dataset with information on height and weight for a group of people. By calculating the eigenvectors and eigenvalues of this dataset, we can:
Find the direction in which height and weight are most correlated (an eigenvector showing the pattern).
See how important this correlation is with an eigenvalue, which tells us if this pattern captures a big or small portion of the data's overall "shape."
God Tier Explanation
I think it's important to point out what an operator can do to a vector (A*x) in general, and then point out that these eigen directions are special, because here the operator's effect is just scaling. And this is useful, because...
This is amazing. You are amazing.
thanks!
This is amazing! Thank you so much for making these videos.
WOW!! bless you man🌺
Lovely! I have a question: if the scalar (eigenvalue) is negative, when multiplied is the vector's direction not changed by 180 degrees?
Killed it. Period.
Explanation is awesome
Great video!
Thank you for this great explanation .
Glad it was helpful!
Take the last section of the video, knowing that eigen is a German word for "own" and you will never forget what is the importance of eigenvalues and eigenvectors.
Thank you, really helpful, awesome explanation :)
Hey, special thanks for that last application example 😊
Thanks man! Really good explanation! Keep it up!
Thanks, will do!
really nice explanation thank you!!
just found you & I LOVE YOU
thank you I love this topic it gives me a lot in the foundation of basic data science most likely in machine learning
These videos are amazing!
This helps a lot!! Thank you so much
Thanks for your clear explanation
Good explanation of the math. But for 40 years I still struggle with what eigenvalues really are. Your fish example was better than most I have heard but I am still missing something vital.
It's definitely a tricky concept and I'm glad this video helped a little bit. Took me a long time to understand too. I think the easiest explanation is that an eigenvector is one where the matrix will map a vector to a multiple of itself (so that the input vector and the output vector both point in the same direction). Why does this matter? Because the same direction ensures the same ratios between each individual vector component which loosely means that the input and output vectors have the same proportions.
@@ritvikmath That helps. I have come across it in PCA and in quantum mechanics. (Yes, I am eclectic).
Another question: do eigenvalues HAVE to be real?
How did u assume the shape of X was (2,1)?
Was it because of two eigen values?
2:51 Thank you. A teacher who melds big pictures with equations
great job
How does this property of a vector (eigen vector) remains in the same dimension even after transformation (by A) helps in some problem solving (related to ML)?
Thank you very well explained and I like the fish analogy. .
All teachers seem to fail at the stage where they include the identity matrix [02:07] why is that? The reason is, because they understand that putting the identity matrix in does not affect vector or lambda. But they never tell you this vital bit of information. And they still wonder why people fail to understand mathematics. They had to learn it themselves. but they are not including this vital piece of knowledge in their explanations. It now seems a trivial point to them, but for a student starting out, it is not trivial. In fact, for any student with a basic understanding of Algebra, they would wonder why I only one side of the equation Ax = λx being multiplied by I, the identity matrix, surely this breaks the rule of algebra which says whatever you multiply one side of the equation by, you must multiply the other side of the equation by. And yet here, we see only one side of the equation being multiplied by I, the identity matrix. Without any explanation as to why you can do that. It's time you start explaining why it is alright to multiply λx by I, the identity matrix on one side and not the other side of the equation: answer, because as any identity does, it does not change the number. hence the word identity. [02:23] let's subtract, Ax - λx = λIx, ready? Ax - λx -[λIx]= λIx - [λIx], okay how does that equal Ax - λIx = 0? well, λIx - [ λIx] equals zero, so the right hand side of the equation is fine. but what about the left hand side? What we have is Ax - λx -[λIx] . Okay so let's apply a little algebra: like terms can be added or subtracted. No like terms so, nothing can be subtracted here. Amazingly, ritvikmath seems to think these can be subtracted. Actually, in his calculations the term λx just magically disappers, so he is left with Ax - λIx = 0. He could have got to this result a different way.
Let's start out with Ax = λx, then subtract λx from both sides (as laws of algebra suggest) that would give: Ax - λx = λx - λx. which results in, Ax - λx = 0. Now he has a choice to include the identity matrix Ax - λIx = 0. see, same result. Nothing magical, nothing disappears, every step accounted for. His main argument is right. And I look forward to his video on determinants of matrices, i.e. proving that a matrix is non-invertable.
funny how the mathematical understanding behind it is very important to grasp, however we will never have to calculate the eigenvectors and values by hand after university.
Good video! Thanks.
great explanation!!
You are good 👍
I have a quick question, at 1:01 you mention that lambda is such that it is a real number, can't this be extended to imaginary numbers as well? Btw, Thanks for you great work!
Yes.
amazing !!
Could you upload a video on Hat Matrix?
In aerodrums screens eigen vector and eigenvalues for different landing planes may be manipulated with out collision by having graphics accordingly correct? That might have been a better explanation.
my man gave the fish a mohawk :))) thanks for the content though. much love
Does matrix A have to be square?
Thank you
soooo good！
Fantastic
Evals and Evecs are everywhere in DS
I just subscribed!
yay! thanks :)
thank you soooo much
You're welcome!
so we always gonna use only one eigenvalues, am I right?
great fish !
haha thanks!
Talking about fishy vectors... 8:31
Math and engineering classes always seem to treat Ax = λx as an abstraction. I wish someone would say at the beginning of the discussion that Ax = λx means that an eigenvector is a vector that points in the same direction after it's been operated on by A.
they could point in different direction though, is lamda is a negative number
@@s25412 My comment about direction was a generality. Of course, A might transform x so that it points in the opposite direction, but the eigenvector will point along the same line as it was pointing before being operated on by A. A scalar multiple of an eigenvector is also an eigenvector.
@@robertpenoyer9998 Thank you
hey, can we subtract the mean of each column from the column so as to make it zero mean before calculating the cov matrix. and in some textbooks it is divided by n-1 instead of n. why is that? Thanks
I think because with (n-1) the estimator is unbiased...
It is because of the difference between "population" and "sample" if you use for population then the accuracy must be considered so that we use n-1 it's for more accuracy.
Didn't really go over finding the Eigenvector 😕 just solved the system of equations and left it be.
does Ax = LAMDA X holds for all x?
Nope
新たに定理を発見しました。
Why couldn’t you have been my teacher when I was studying eigenvectors. Sigh.
Hi, could you please do the computation for eigen value -2 and eager to know how to plug in x1 and x2
🌹🌹🌹
🌹 🌹 🌹
Wondering who hit dislike!!
I'm crying
Actually, the concepts are foundational....