Absolutely, absolutely excellent! The visuals; the clear, thorough explanations; the excellent balance of example and theory: they all came together to make a very informative video that increased my intuition of Linear Algebra. Thank you for this video! I'm so glad I watched this.
i think this is important for PCA (Principle Component Analysis ) and i am super thankful for getting a intuitive visualization on this topic instead of having to memorize equations and proofs.. THANK YOU!
Heyo! Just wanted to tell you that you've absolutely saved my life. I was really struggling with the connection between diagonalization and eigenvalues, and this video turned it from a needlessly theoretical concept, into feeling almost obvious intuitively. Thanks a bunch!
Holy cow you and your videos are amazing….in in uni know and discovered your channel because of Linear Algebra, and I have to say your content is wonderfully insightful and just EASY TO UNDERSTAND! Your online book is also amazing, you should be proud because you are carrying future generations of physicists, mathematicians and engineers. If there is anywhere we could make a donation, I’d be happy 😊. THANK YOU ♥️
You are right up there with 3b1b in regards to excellent math content. Keep up the good work 😃 this is the time for mathematicians to shine and make videos explains all topics and building intuition. I really love this blow up of videos on UA-cam, and I am subscribing to lots of math channels. I am truly forever grateful for your work. I say this with all my heart, thank you.
The visuals at the end were just what I needed. I'd been struggling to understand why it was P^-1 that converted from the standard basis instead of P, the visuals helped me actually see what was happening. All around excellent video! Also this is the second time today that I stumbled across this channel (the first was related to Calc 3 and vector fields)
Thanks for the content. I think this way, I can understand much more because I can see how the eigenvectors and eigenvalues are important to linear transformation. Visualizing the process turn it easier to learn.
WOW this was so helpful and exactly what i needed!! i somehow made it through two whole linear algebra classes without actually understanding the meaning of PDP^(-1) decomposition, struggling with the computations because i didn't really understand what i was doing. this video truly was a lightbulb moment for me, thank you so much!
man i wish you had more views, just sad, because the video is great. Professor Strang and his mit lectures are cool but they lack visualizations. Thank you very much
Bravo, Dr. Bazett!! This is a fantastic video and you have presented the materials elegantly. Thank you so much for your contributions to the world of Mathematics, sir! :-)
it takes the idea of change of basis. Nice. Like me, I get used to use orthogonal basis in my language. Anything different from my orthogonal, I would give it a transform to my orthogonal basis first. thank you for your video interpretation about this idea.
Can you please give me a clue on how to do that grid animations, I need to give a lecture to collage students, and showing such intuitive animations will add great value to their understanding.
I think im confused on what you mean bu eigenbasis and standard basis. I thought the standard basis would be the cartesian plane that we’re used to… i dont understand
Isn't the matrix [1,-1;1,1] converting to the eigenbasis from the standard basis? (because those are the eigenvectors and the standard basis would just be [1, 0; 0 1])?
that tripped me up as well but rewatch and remember red arrow is always (1, 1) and yellow arrow is always (-1,1). Applying P^(-1) has shifted these to (1,0),(0,1); we are now in the perspective of the people whose standard basis is (1,1) and (-1,1). To them they don't write it diagnonally but straight up and across
Awesome as usual. But I'm confused here to get clear idea about eigen basis and real standard grid.By the way suppose I have a matrix and I find all eigen value and eigen vector, now how can I use those to make my original matrix easier geometrically?How can I form a easier or suitable coordinate grid by eigen vectors?And please explain What is "principle" direction. Thanks again.
How come P takes vectors in the eigen basis and expresses them in the standard basis? Surely it should be the other way around because the standard basis gets mapped onto the basis in the matrix by the definition of matrix multiplication sorry I'm just confused here :)
@@DrTrefor I think there is a mix of terminologies here. I think you refer to the standard basis as the basis you are transforming to rather than the usual standard coordinate basis, am I right? Am confused with the explanation too!
P^(-1)*x transforms a vector x such that it is given in terms of the eigenvectors (change of basis from standard to eigen). With this choice of coordinates we know that the transformation is applied by simply scaling the vector P^(-1)*x with scaling matrix D. However, the scaled vector D*P^(-1)*x is still expressed in terms of the eigenvectors and has to be transformed back to standard coordinates with P (change of basis from eigen to standard). This video is overall a clear and concise explanation of the concept behind Eigendecomposition.
hey dr. u made a minor error in reporting the eigenvector and eigenvalue It should have[-1,1] on both LHS and RHS knowing that this video is more than 4 years old you must have already noticed it
I love the content of this channel but the audio is so bad that I just can’t listen to it. I do t think I’ve made it through a single video with the sound on. Your continent is so good please please please invest in a better mic and something to absorb the echo and a little bit of post processing with sound. With your intonation and voice it is lost in the sound
Here's my try as a non-expert. We started by trying to find the diagonal matrix D such that AP= PD. In other words, we want to know what diagonal matrix D acts on P in the same way that A transforms P. Since, by definition, an eigenvector P of A is a vector where multiplication by A just stretches P in the same direction (which is another way of saying that it just multiplies P by some set of scalars). After we find eigenvalues and eigenvectors for A, then we can recognize that another way of scaling P by some constants is to multiply a diagonal matrix with those scalars on the diagonal by P. So we write down our diagonal D of eigenvalues, then multiply by the eigenbasis P. The purpose of this is to represent A in a convenient way, so we just multiply by P^-1 to isolate A. Then, A = PD(P^-1).
By the definition of Eigenvector and Eigenvalue (used 2d example) : A*p_1 = p_1*d_1 A*p_2 = p_2*d_2 Where d is eigenvalue (which can be converted into diagonal matrix D), and p is the eigenvector, and A is the original matrix. Put p_1 and p_2 into matrix P A*P = P*D multiply P by P^-1 to get A (must be on the right side of P because matrix multiplication is not communitative) A = PDP^(-1)
4:43 there is a type at the right top corner: the eigenvector should be (-1,1) instead of (1,1)
comment there is a typo in your fourth word: the word should be 'typo' instead of 'type'
Thank you!
@@DriggerGT3 english is not math
Absolutely, absolutely excellent! The visuals; the clear, thorough explanations; the excellent balance of example and theory: they all came together to make a very informative video that increased my intuition of Linear Algebra. Thank you for this video! I'm so glad I watched this.
Thank you so much! The visualisation not only makes the concept clear but also helps in the intuition. Brilliant!
i think this is important for PCA (Principle Component Analysis ) and i am super thankful for getting a intuitive visualization on this topic instead of having to memorize equations and proofs.. THANK YOU!
Heyo! Just wanted to tell you that you've absolutely saved my life. I was really struggling with the connection between diagonalization and eigenvalues, and this video turned it from a needlessly theoretical concept, into feeling almost obvious intuitively. Thanks a bunch!
Thank you for making linear algebra more Fun and visualizable
Holy cow you and your videos are amazing….in in uni know and discovered your channel because of Linear Algebra, and I have to say your content is wonderfully insightful and just EASY TO UNDERSTAND! Your online book is also amazing, you should be proud because you are carrying future generations of physicists, mathematicians and engineers. If there is anywhere we could make a donation, I’d be happy 😊. THANK YOU ♥️
Damn man! You are really underrated. You should have way more subscribers.
You are right up there with 3b1b in regards to excellent math content. Keep up the good work 😃 this is the time for mathematicians to shine and make videos explains all topics and building intuition. I really love this blow up of videos on UA-cam, and I am subscribing to lots of math channels. I am truly forever grateful for your work. I say this with all my heart, thank you.
The visuals at the end were just what I needed. I'd been struggling to understand why it was P^-1 that converted from the standard basis instead of P, the visuals helped me actually see what was happening. All around excellent video! Also this is the second time today that I stumbled across this channel (the first was related to Calc 3 and vector fields)
Thanks for the content.
I think this way, I can understand much more because I can see how the eigenvectors and eigenvalues are
important to linear transformation.
Visualizing the process turn it easier to learn.
clear and beautiful. my brain still refuses to see the mechanics behind all this, keeps taking me back to standard canonical brain.
WOW this was so helpful and exactly what i needed!! i somehow made it through two whole linear algebra classes without actually understanding the meaning of PDP^(-1) decomposition, struggling with the computations because i didn't really understand what i was doing. this video truly was a lightbulb moment for me, thank you so much!
Thank you Dr. Trefor, you are a genius.
You are the gold standard for education content
man i wish you had more views, just sad, because the video is great. Professor Strang and his mit lectures are cool but they lack visualizations. Thank you very much
The visuals in the last couple minutes are a nice touch.
Thank you so much for this fantastic video! Your videos are incredible, I've been binging them on and off the last few months.
Truly amazing video! Great visualisations and graphics, Thank you!
Bravo, Dr. Bazett!! This is a fantastic video and you have presented the materials elegantly. Thank you so much for your contributions to the world of Mathematics, sir! :-)
You’re most welcome!
man your the only reason i am getting through this course. Thank you
Now i have found something than 3blue1brown...
excellent visual explanation....
Thank you
Love the enthusiasm. Great video.
Thank you so so much!!! A very simple explanation of a complex topic))
Coincidental that the "How the Diagonalization Process Works" video was posted exactly a year back
Thank you for the visual comparison between the two basis.
Thank you, Trefor for once again showing me the way. 🤩🤩🤩
P.S. Is this part of a series? I don't see links to it, if that's the case.
Thanks! Part of my intro to linear algebra playlist, should be findable on the homepage.
Extremely helpful and clearly explained Thanks!
Thank you ..your videos are so nice....wow we really appreciate you...❤️
Very clear explanation!thank you.
Simply amazing and extremelly helpful video! You're the best :)
it takes the idea of change of basis. Nice. Like me, I get used to use orthogonal basis in my language. Anything different from my orthogonal, I would give it a transform to my orthogonal basis first. thank you for your video interpretation about this idea.
Thanku youuu sir.. Thank you very much.. really really greatful.. Please keep trating us with such visual and conceptual treats!
Thanks for the video. Seems that you gave me intuitive kick.
Great video. Thank you!
Whoa!!! I understand it now!!!
This was such an excellent insight, thanks a lot!
🤩🤩🤩🤩 Amazing explanation
Fantastic. I learned so much in no time
U deserve million views
Can you please give me a clue on how to do that grid animations, I need to give a lecture to collage students, and showing such intuitive animations will add great value to their understanding.
I think im confused on what you mean bu eigenbasis and standard basis. I thought the standard basis would be the cartesian plane that we’re used to… i dont understand
Could you visualize how A= pdp-1? Thanks for showing how the matrix changes between pdp-1, though!
Essentially, how does sneaking in a diagonal effect the end result
(Show P, then PD, then PDP-1
Isn't the matrix [1,-1;1,1] converting to the eigenbasis from the standard basis? (because those are the eigenvectors and the standard basis would just be [1, 0; 0 1])?
I have the same dount. Can someone explain that ?
For understanding this, go through these videos in his Linear Algebra playlist :
1. Changing between basis
2. Visualizing change of bases dynamically
its matter of prespective what he is saying that [c1 c2] are written in eigen basis to us which is standard basis
that tripped me up as well but rewatch and remember red arrow is always (1, 1) and yellow arrow is always (-1,1). Applying P^(-1) has shifted these to (1,0),(0,1); we are now in the perspective of the people whose standard basis is (1,1) and (-1,1). To them they don't write it diagnonally but straight up and across
Awesome as usual. But I'm confused here to get clear idea about eigen basis and real standard grid.By the way suppose I have a matrix and I find all eigen value and eigen vector, now how can I use those to make my original matrix easier geometrically?How can I form a easier or suitable coordinate grid by eigen vectors?And please explain What is "principle" direction. Thanks again.
Trés bien expliqué
merci bien
Big thanks from korea, helped a lot
very clear and intuitive explanation!!
Which software is it?? Very very nice explanation. Greetings from italy!
wow this is brilliant !! thanks a ton
Really good❤️
So cool, thank you so much
You're most welcome!
Thanks a lot !!! How to diagonalize a rotational matrix?
Great vid! Thank you :)
How come P takes vectors in the eigen basis and expresses them in the standard basis? Surely it should be the other way around because the standard basis gets mapped onto the basis in the matrix by the definition of matrix multiplication sorry I'm just confused here :)
@@DrTrefor I think the confusing-ness of this is actually due to the existence of duality perhaps? :) thank you for your quick response man:D
@@DrTrefor I think there is a mix of terminologies here. I think you refer to the standard basis as the basis you are transforming to rather than the usual standard coordinate basis, am I right?
Am confused with the explanation too!
@@chinedueleh3045 No, he isn't
P^(-1)*x transforms a vector x such that it is given in terms of the eigenvectors (change of basis from standard to eigen). With this choice of coordinates we know that the transformation is applied by simply scaling the vector P^(-1)*x with scaling matrix D. However, the scaled vector D*P^(-1)*x is still expressed in terms of the eigenvectors and has to be transformed back to standard coordinates with P (change of basis from eigen to standard).
This video is overall a clear and concise explanation of the concept behind Eigendecomposition.
@Dr. Trefor Bazett I have the same dount. Can someone explain that ?
Brilliant !!! Thank you !!!
nice video!
Thanks!
thank you so much!
very helpful thanks
very good with thx
Here in case of diagonalization geometrically and numerical different meaning?
Life saver
After watching your video on how to diagonalize a matrix I immediately looked for a visual explanation and what do I find?
haha so important to have that visual piece!
not all heroes wear capes
How is the eigenvector both (1 1) and (-1 1)?
At @3:18
legend
hey dr. u made a minor error in reporting the eigenvector and eigenvalue It should have[-1,1] on both LHS and RHS knowing that this video is more than 4 years old you must have already noticed it
He sounds like 3blue1brown.👀
What are the Cs at 6:38?
s3x is nice but have you heard of diagonal matrices?
I love the content of this channel but the audio is so bad that I just can’t listen to it. I do t think I’ve made it through a single video with the sound on. Your continent is so good please please please invest in a better mic and something to absorb the echo and a little bit of post processing with sound. With your intonation and voice it is lost in the sound
I swear I will follow and share your videos once you fix the sound on them.
I still don't understand it
Sir, Please upload full course of linear and abstract algebra
who's here because they took 211 with the wrong prof
haha, 211 at uvic?
@@DrTrefor :)
Where did you get the P^-1 from?, and please make it easy to understand it not like you understand it.
Thank you
Here's my try as a non-expert. We started by trying to find the diagonal matrix D such that AP= PD. In other words, we want to know what diagonal matrix D acts on P in the same way that A transforms P. Since, by definition, an eigenvector P of A is a vector where multiplication by A just stretches P in the same direction (which is another way of saying that it just multiplies P by some set of scalars). After we find eigenvalues and eigenvectors for A, then we can recognize that another way of scaling P by some constants is to multiply a diagonal matrix with those scalars on the diagonal by P. So we write down our diagonal D of eigenvalues, then multiply by the eigenbasis P. The purpose of this is to represent A in a convenient way, so we just multiply by P^-1 to isolate A. Then, A = PD(P^-1).
By the definition of Eigenvector and Eigenvalue (used 2d example) :
A*p_1 = p_1*d_1
A*p_2 = p_2*d_2
Where d is eigenvalue (which can be converted into diagonal matrix D), and p is the eigenvector, and A is the original matrix.
Put p_1 and p_2 into matrix P
A*P = P*D
multiply P by P^-1 to get A (must be on the right side of P because matrix multiplication is not communitative)
A = PDP^(-1)
You
Great video, thank you!