3:20 Those of you confused about the free variables: The first column has a leading 1 that corresponds to x1, so it's not a free variable. Neither of second or the third column have a leading 1 "at the last row", that correspond to x2 and x3 respectively, so they are both free variables. So, x2 = s, x3 = t Now, first row: x1 + x2 + x3 = 0 + 0 + 0; Second row: x1 + x3 = x1 + t = 0, we get x1 = -t; Yeet the third row straight away. x1 = -t, x2 = s, x3 = t This general solution can be expressed in a vector form: ( x1 ) ( -t ) ( x2 ) = ( s ) ( x3 ) ( t ) You can decompose this as a linear combination like this (as shown in the video): ( x1 ) ( 0 ) ( -1 ) ( x2 ) = ( s ) ( 1 ) + t ( 0 ) ( x3 ) ( 0 ) ( 1 ) If you sum it up, you get the one I showed above.
Hey Trefor. I've been watching your videos for quite some time now and I JUST realized that you're a professor at my University. Anyway great work. I hope be in your class for calc 3/4.
Great video! Btw just a tip for people: When you have found P and D you can check if AP = PD instead of figuring out P inverse. If AP = PD then you've found the correct values.
Always here on the day before the exam! And I'm never disappointed!! Last semester; for Differential Equations... Now for Linear Algebra!! Thanks Doc!! Respectful!
Thank you for being so clear and engaging. You have interesting/challenging points, you don't shy away from repeating the things you have already covered again and you are making really clear what is the convention and what is something the textbooks do to save some space. I think you just saved me hours of work searching for information!
Hi, I'd like to ask for clarification regarding finding the eigenvectors, with respect to the values t and s. In the second row of the matrix with the eigenvalues subbed in, which is 1, 0, 1, setting x3 = t means that x1 = -x3 = -t. But does that not imply that x2 = 0? So why would the column vector of s be (0, 1, 0)? I'm getting a bit confused there.
However, since all the corresponding multipliers for x2 in the matrix are 0 and hance it really doesn’t matter what value x2 takes up. Hence it takes up the span (0,1,0) value
With that S an T, just ignore what he does there, That's gibberish. JUST work out your eigen vectors in descending order, then put them in the modal matrix
What constitutes a column being a free variable is if it has a pivot position, x2 and x3 do not have a pivot position so they are labeled as free variables while x1 has a pivot position making it a basic variables, hope this helps! (Also OneLife in Hagerstown is hiring(Thought you might want to know))
Oh wow, first of all thank you for making this video because without you professor I would have put the pen down whenever I came across this question. Second of all, I don't study maths in English but in French and the video is just self-explanatory. Thank you for your time and have a nice day !
Hey! Could you do a video for identifying conic sections? Like using a diagonalized matrix to the simplify an equation, changing coordinates to then see if its a parabola, ellipse, etc..
Sorry if this is obvious, but why did you do the P calculation at all? If you already knew that the diagonalized matrix had the diagonal equal to the eigenvalues, and you found the eigenvalues super early on, why did you need the eigenvectors?
I don't really understand in which order I put the eigenvalues in D. Is it always ascending order or what's the trick? Also does finding the eigenvectors have any relevance to finding the diagonal matrix D, since you only need the eigenvalues right?
Thank you Sir. Ii just do not understand the theories of diagonalization yet to be able to explain some things but I see that it could also be a PCA dimensionality reduction technique.
Normally, for a matrix to be diagonalisable, it should have distinct eigenvalues but in that case we have two eigenvalues with the same number and another condition to be respected is that the eigenvalues should be lineraly independent as well as the eigenvector. I would be grateful if u could help me
Great video Dr Trefor. I am curious to know the software setup/tools you are using so that you combine your notes with your own self video in front. Thanks in advance for your cooperation.
trefor Bazett + 3blue1 bown essential linear algebra playlist; you will find this subject soooooooo enjoyable and will blow up your mind and develop your visualization skill ALOT; you don't even need these trash lecture in the universities ; they just represent the theoretical side without visualizing perspective which make the topic harder and ungraspable AND boring .
8:47 You're right, I've made a mistake. It's always such a brain-strain for me to spend so much time on one quick example. But if you need to be good at Math you got to accept its jealousy of your time
Late reply, but maybe others are wondering about the same thing. Since it's a 3x3 you need 3 linearly independent eigenvectors. You get one from one of the eigenvalues, and two from the other. If the basis for the eigenspace for the eigenvalue with multiplicity of 2 only produced one, you'd only have 2 vectors which means the 2nd eigenspace collapses one dimension.
Why you took the X3 is the negative value. I.e x3=-t while finding the eigen vector for for eigen value 2.i am little bit confused in that step.. If possible clearly explain...
@@DrTrefor Thanks for your quick response! I watched the video and I totally followed it, however in this example I don't see why you would state that x2 = s, because it seems to me that x2 = 0. In my computation I stated x1 = s and x3 = t, which obviously doesn't lead to the right answer.
Instead of taking (v1 v2 v3), can i take (v3 v2 v1) etc. I do know this will change the diagonalized matrix but does the order of eigen vectors in the mattix really matters?
You know what Dr. Trefor Bazett, I do not like the way that you use s and t as variables for x2 and x3 and makes things much more complicated and confusing for comprehension! I think next time you should just use x2 and x3 as the variables for them while doing the example, also Roy Rogers in downtown Hagerstown is hiring! (Thought you might want to know)
![Diagonalizing 3x3 Matrix - Full Process [Passing Linear Algebra]](http://i.ytimg.com/vi/0kA_VqHSpGg/mqdefault.jpg)
![Diagonalizing 3x3 Matrix - Full Process [Passing Linear Algebra]](/img/tr.png)
3:20 Those of you confused about the free variables:
The first column has a leading 1 that corresponds to x1, so it's not a free variable.
Neither of second or the third column have a leading 1 "at the last row", that correspond to x2 and x3 respectively, so they are both free variables.
So, x2 = s, x3 = t
Now, first row: x1 + x2 + x3 = 0 + 0 + 0;
Second row: x1 + x3 = x1 + t = 0, we get x1 = -t;
Yeet the third row straight away.
x1 = -t, x2 = s, x3 = t
This general solution can be expressed in a vector form:
( x1 ) ( -t )
( x2 ) = ( s )
( x3 ) ( t )
You can decompose this as a linear combination like this (as shown in the video):
( x1 ) ( 0 ) ( -1 )
( x2 ) = ( s ) ( 1 ) + t ( 0 )
( x3 ) ( 0 ) ( 1 )
If you sum it up, you get the one I showed above.
Thanks. Wish I could giv you a fellatio rn😭
Hey Trefor. I've been watching your videos for quite some time now and I JUST realized that you're a professor at my University. Anyway great work. I hope be in your class for calc 3/4.
Great video!
Btw just a tip for people:
When you have found P and D you can check if AP = PD instead of figuring out P inverse.
If AP = PD then you've found the correct values.
Makes since, right multiply both sides by P
Always here on the day before the exam! And I'm never disappointed!! Last semester; for Differential Equations... Now for Linear Algebra!! Thanks Doc!! Respectful!
Good luck!!
1:34 *******dab*******
ua-cam.com/video/D4X1udMDSfw/v-deo.html
Still confused about the free variables
I saw many videos. MANY. This one made SO MUCH SENSE to me. Thank you.
Glad it finally clicked!
Thank you for being so clear and engaging. You have interesting/challenging points, you don't shy away from repeating the things you have already covered again and you are making really clear what is the convention and what is something the textbooks do to save some space.
I think you just saved me hours of work searching for information!
"Because I lie a lot" lmao
You were the best math lecturer I ever had at U of T - I'm still using your videos to help my brother through school now!
DR. Bazett thank you for the video on Diagonalization of a Matrix.
Well explained Dr. Bazett.
Hi, I'd like to ask for clarification regarding finding the eigenvectors, with respect to the values t and s.
In the second row of the matrix with the eigenvalues subbed in, which is 1, 0, 1, setting x3 = t means that x1 = -x3 = -t. But does that not imply that x2 = 0? So why would the column vector of s be (0, 1, 0)? I'm getting a bit confused there.
Yeah I think so too
However, since all the corresponding multipliers for x2 in the matrix are 0 and hance it really doesn’t matter what value x2 takes up. Hence it takes up the span (0,1,0) value
With that S an T, just ignore what he does there, That's gibberish. JUST work out your eigen vectors in descending order, then put them in the modal matrix
Finally, a video neither skip x calculation nor keep mention how to calc with known P : ) Thank you sir
so clear, ty ♥
My exam is in 2h. Wish me luck.
Haha didn’t get this until 3 hours latter but...uh....good luck!
hi Trefor, quick ask. what conditions constitute choosing a free variable like you did for x2 and x3?
What constitutes a column being a free variable is if it has a pivot position, x2 and x3 do not have a pivot position so they are labeled as free variables while x1 has a pivot position making it a basic variables, hope this helps! (Also OneLife in Hagerstown is hiring(Thought you might want to know))
Crystal clear presentation, thanks
Amazing Explanation
Thank you, you're a legend.
Oh wow, first of all thank you for making this video because without you professor I would have put the pen down whenever I came across this question. Second of all, I don't study maths in English but in French and the video is just self-explanatory. Thank you for your time and have a nice day !
Thank you! This is very helpful and included a lot of points in one vid and was all to the point!
Pretty good explanation honestly.
greeting from France you are helping students at the international level !
I am so.... tired. Finals week. Thank you for your help Trefor🙏🏻
Crystal clear 👌
Well explained 🤗
Hey! Could you do a video for identifying conic sections? Like using a diagonalized matrix to the simplify an equation, changing coordinates to then see if its a parabola, ellipse, etc..
Excellent teaching.... Thanks a lot Sir ji 👍
Most welcome
may i know why the free variables are x2 and x3?
Sir
Your presentation is very unique and very good
It help me lot
This was a very informative video, Thanks!
Omg my final is tomorrow and this literally saved me
Thank you! This is so much more clear than my professor.
Find the matrix P that diagonalizes A.
Where A= 100120-352 . Are the eigen vectors linearly independent?
Sorry if this is obvious, but why did you do the P calculation at all? If you already knew that the diagonalized matrix had the diagonal equal to the eigenvalues, and you found the eigenvalues super early on, why did you need the eigenvectors?
Great explanation of the concepts. Really simple and so easy to follow. Thanks!
Very well explained, thank you.
love this video!! explain things so simple that is easy for me to understand!
I don't really understand in which order I put the eigenvalues in D. Is it always ascending order or what's the trick? Also does finding the eigenvectors have any relevance to finding the diagonal matrix D, since you only need the eigenvalues right?
I am gonna have this problem on my final. this is a great help
thank you. It really helped.
This is super helpful, thank you so much!
Thanks a lot from Brazil.
I need to learn this subject because next week I'll have a linear algebra exam.
Excelent explanation.
how was x2,x3,x1 found ?
and how was v3 found ?
Thank you
Thank you.Helped me lot.
Thank you Sir. Ii just do not understand the theories of diagonalization yet to be able to explain some things but I see that it could also be a PCA dimensionality reduction technique.
Thanks! Helped a lot! :)
Thank you Finally Get this
you are definitely better than my teacher... ♥♥♥♥♥♥
Wonderful sir, the best one that I have watched ever.
Thank you very much!!!!!! Helped a lot.
Thank you so much, well explained
Diagonalization is not on the form PAP-1, instead it is P-1AP. Please verify and make the suitable changes.
True, but I said A=PDP^-1 which is equivalent to what you said. I think you confused the A and the D
@@DrTrefor Thanks for the clarification, I also realized that later after raising the question.
thanks you sir ,your explanation inspiration for me
So nice of you!
Sir i have a question how you assigned values to x1,x2 and x3 . I did not got that.
Normally, for a matrix to be diagonalisable, it should have distinct eigenvalues but in that case we have two eigenvalues with the same number and another condition to be respected is that the eigenvalues should be lineraly independent as well as the eigenvector. I would be grateful if u could help me
3:15 whats that. I didn't get it
Give this man a Victoria Cross
Great video Dr Trefor. I am curious to know the software setup/tools you are using so that you combine your notes with your own self video in front. Thanks in advance for your cooperation.
Thank u sir😎
trefor Bazett + 3blue1 bown essential linear algebra playlist; you will find this subject soooooooo enjoyable and will blow up your mind and develop your visualization skill ALOT; you don't even need these trash lecture in the universities ; they just represent the theoretical side without visualizing perspective which make the topic harder and ungraspable AND boring .
can we just use elementary row and column transformations to achieve the same result?
Nice video Prof. Please tell me technology you are using to make such videos
Just writing on my tablet with a greenscreen behind me
@@DrTrefor very nice.
What is the way to mask the background green screen.
Thank you teacher
Awesome videos. Very clear!
I was taught that the determinant is (lamda I - A). Is this wrong or does it not matter?
Just love it 😍
8:47 You're right, I've made a mistake.
It's always such a brain-strain for me to spend so much time on one quick example.
But if you need to be good at Math you got to accept its jealousy of your time
Hi I have a question, does in diagonalization, one or two of the x has to be zero so that we can put arbitrary values? Thank You
But is A really diagonalizable when you have a (λ-2)^2? Shouldn't the eigenvalues be distinct or have I missed something?
Late reply, but maybe others are wondering about the same thing. Since it's a 3x3 you need 3 linearly independent eigenvectors. You get one from one of the eigenvalues, and two from the other. If the basis for the eigenspace for the eigenvalue with multiplicity of 2 only produced one, you'd only have 2 vectors which means the 2nd eigenspace collapses one dimension.
Eigenvectors from different eigenvalues are linearly independent but eigenvectors from the same eigenvalue are not inherently independent, no?
Can I do row operations to make the matrix be a triangular matrix and then calculate the eigenvalues?
So is the diagonalised matrix D, that which has the same information as A but in a different basis (the eigenbasis?)
Why you took the X3 is the negative value. I.e x3=-t while finding the eigen vector for for eigen value 2.i am little bit confused in that step.. If possible clearly explain...
can anyone explain how he goes from the system to the notation with s and t? Around 3:20
@@DrTrefor Thanks for your quick response! I watched the video and I totally followed it, however in this example I don't see why you would state that x2 = s, because it seems to me that x2 = 0. In my computation I stated x1 = s and x3 = t, which obviously doesn't lead to the right answer.
@@celine80357 im having the same questions, since x2 is all zeros column and x1 and x3 have 1's in the middle of the column
Instead of taking (v1 v2 v3), can i take (v3 v2 v1) etc. I do know this will change the diagonalized matrix but does the order of eigen vectors in the mattix really matters?
When calculationg values. Why not (2-lambda)^2*(1-lambda) - (2-lambda)?
thank you!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
I Really don't understand at 3:12 to determine the free variables and then showing what column to use
it doesnt matter that i take lambda1 =1, lambda2 =2 right?
It's very interesting 🙏
beautiful
Who else got this in 2024😂
You computed the eigenvalues wrong right? It is supposed to be det(λI - A) not det(A - λI)
You know what Dr. Trefor Bazett, I do not like the way that you use s and t as variables for x2 and x3 and makes things much more complicated and confusing for comprehension! I think next time you should just use x2 and x3 as the variables for them while doing the example, also Roy Rogers in downtown Hagerstown is hiring! (Thought you might want to know)
When I multipled PDP^(-1) , it didn't equal A 😐
4:36 now i can understand
I have multiplied but I can not see A = PDP(-1). has anyone tried to compute A? I have taken A, P and D from given example.
hi why is x3 a free variable in 3:26
thank ya
Learning sth from non indian guy. That's the dream
what's sth
@@motherisapesth means something
@@loogoos4894 ok
That sounds so racist
@@jackblurton9153 it is
I didn't quite get where you got v1, v2, v3. Anybody have any insight?
why is x2 's'? Why is it not zero?
I have done matrix multiplication. PDP-1 is not equal to A. anyone has tried to do that?
Thnaks
thanks
thanks :-)
don't you have to normalize your eigenvectors?
thank you so much