Full Example: Diagonalizing a Matrix

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Let's compute a full example of Diagonalizing a matrix via eigenvectors and eigenvalues. The steps:
1) Compute the eigenvalues. Take the determinant of A-lambda I and set it equal to zero. The solutions are the eigenvalues
2) Compute the eigenvectors. For each eigenvalue solve (A-lambda I)x=0, which is a homogeneous system
3) Form the P and D matrix. The D matrix is just the eigenvalues down the main diagonal. The P matrix is the corresponding linearly independent eigenvectors for each eigenvalue as columns, in the same order as their eigenvalues were put into D.
4) Finally, we can test that A=PDP^-1 is true by multiplying out the right hand side.
How to VISUALIZE the diagonalization process: • Visualizing Diagonaliz...
How to COMPUTE the diagonalization proces: • How the Diagonalizatio...
Learning Objective: Starting with a matrix, compute the eigenvalues and eigenvectors and then finally diagonalize the matrix.
Course Playlists:
►Linear Algebra: • Linear Algebra (Full C...
►Calculus I: • Calculus I (Limits, De...
►Calculus II: • Older Calculus II (New...
►Discrete Math: • Discrete Math (Full Co...
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►Follow me on Twitter: / treforbazett
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This video is part of a Linear Algebra course created by Dr. Trefor Bazett

[@TheRagingSun]

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.

[@simenl1289]

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.

[@MinthZe]

Makes since, right multiply both sides by P

[@fr0styy202]

1:34 *******dab*******

[@SURYAKOURAVVIENS]

kzbin.info/www/bejne/emW7YqiagqmGnNk

[@tonystank1309]

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!

[@DrTrefor]

Good luck!!

[@topchrischang471]

love this video!! explain things so simple that is easy for me to understand!

[@lebalen]

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..

[@audreakimsley9991]

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

[@Trisevic]

Great explanation of the concepts. Really simple and so easy to follow. Thanks!

[@pranavmahapatra]

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.

[@RazorIance]

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.

[@abhimanyutimbadia8467]

Yeah I think so too

[@abhimanyutimbadia8467]

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

[@JPKB36]

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

[@peienshao3936]

Finally, a video neither skip x calculation nor keep mention how to calc with known P : ) Thank you sir

[@lowisl289]

may i know why the free variables are x2 and x3?

[@panyasilesiba4848]

hi Trefor, quick ask. what conditions constitute choosing a free variable like you did for x2 and x3?

[@ethanclutter880]

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))

[@stephlovesmusic3167]

Thank you! This is very helpful and included a lot of points in one vid and was all to the point!

[@alkendimacale8047]

Can someone explain the eigenvector part of this? Like after the row reduction. thanksss

[@umgubularslashkilter9272]

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!

[@ricki7389]

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?

[@umairfarooq5862]

Sir i have a question how you assigned values to x1,x2 and x3 . I did not got that.

[@BrutalOddball]

I didn't quite get where you got v1, v2, v3. Anybody have any insight?

[@MostafaAbdelrehim]

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.

[@jliu7221]

Can I do row operations to make the matrix be a triangular matrix and then calculate the eigenvalues?

[@Brantendo64]

I saw many videos. MANY. This one made SO MUCH SENSE to me. Thank you.

[@DrTrefor]

Glad it finally clicked!

[@user-hf1yv3nx6d]

Wonderful sir, the best one that I have watched ever.

[@anup_narvekar]

Very well explained, thank you.

[@gabrielpereiramendes3463]

Thanks a lot from Brazil. I need to learn this subject because next week I'll have a linear algebra exam. Excelent explanation.

[@clementchan8544]

How do you know which lamda value will be 1/2/3 or v matrix be 1/2/3? For example if i use your lamda3 to find the v matrix first, the P obtained will be quite different from the results that you obtained

[@spi.ritu.ality007]

Trefor Bazett Thanks

[@brycematthews4303]

Thank you, you're a legend.

[@josevela9018]

I Really don't understand at 3:12 to determine the free variables and then showing what column to use

[@achalcharantimath5603]

how was x2,x3,x1 found ? and how was v3 found ? Thank you

[@mohammednaved1973]

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?

[@pashimp]

Thanks! Helped a lot! :)

[@priyankasapali5736]

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...

[@wattyven]

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!

[@nianbing5528]

Pretty good explanation honestly.

[@DiyaeDiyae-hi7cn]

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 !

[@kirtipandya4618]

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.

[@quanttube]

Well explained Dr. Bazett.

[@ryanabel9871]

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

[@serkalemadugna4675]

what condition is diagonal matrix invertible

[@prestonduffield5393]

Crystal clear presentation, thanks

[@mathiasstrand1254]

But is A really diagonalizable when you have a (λ-2)^2? Shouldn't the eigenvalues be distinct or have I missed something?

[@HenrikMonsen]

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.

[@cinnamonshake45]

can we just use elementary row and column transformations to achieve the same result?

[@meatychunkz8875]

So is the diagonalised matrix D, that which has the same information as A but in a different basis (the eigenbasis?)

[@krittaprottangkittikun7740]

This is super helpful, thank you so much!

[@tosinadekunle646]

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.

[@TwistedGamersHQ]

When calculationg values. Why not (2-lambda)^2*(1-lambda) - (2-lambda)?

[@georgesadler7830]

DR. Bazett thank you for the video on Diagonalization of a Matrix.

[@elfabri666]

so clear, ty ♥

[@sweetyranii]

Crystal clear 👌 Well explained 🤗

[@MrBagga-md4mj]

thank you. It really helped.

[@aminaalsindi56]

Thank you so much, well explained

[@alimohmed2578]

"Because I lie a lot" lmao

[@nathanschwedock6338]

I was taught that the determinant is (lamda I - A). Is this wrong or does it not matter?

[@daxtonbryce2712]

Awesome videos. Very clear!

[@AnotherCasualViewer]

This was a very informative video, Thanks!

[@nathanellul5787]

Helped a lot! Thank you

[@therattleinthebook397]

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?

[@vaggelismanousakis6147]

Is it possible in an exercise to find 2 eigenvectors for each eigenvalue so we end up with 6 eigenvectors given a 3x3 Matrix? Please help! Thanks:)

[@vaggelismanousakis6147]

@@DrTrefor Thanks a lot! Have a nice day :)

[@habibnurmagovedov9260]

greeting from France you are helping students at the international level !

[@kirtipandya4618]

I have done matrix multiplication. PDP-1 is not equal to A. anyone has tried to do that?

[@subhanonbera4921]

Sir Your presentation is very unique and very good It help me lot

[@chenzhuo4043]

Thank you very much!!!!!! Helped a lot.

[@therandomactivist1124]

Still confused about the free variables

[@azhua123]

@2:34 what is going on here? how'd you get that matrix?

[@TheOneMusketeer]

Not sure if you still need help but he's basically just subtracting the eigenvalues from the A matrix.

[@MaxWell-ch4jf]

Thank you Finally Get this

[@syremusic_]

I am so.... tired. Finals week. Thank you for your help Trefor🙏🏻

[@celine80357]

can anyone explain how he goes from the system to the notation with s and t? Around 3:20

[@celine80357]

@@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.

[@fatimaguerra6615]

@@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

[@user-rb4bv8st5q]

thank you so much

[@sicaw4259]

it doesnt matter that i take lambda1 =1, lambda2 =2 right?

[@ruwanishiromala]

Thank you.Helped me lot.

[@maureensikora9012]

Thank you! This is so much more clear than my professor.

[@sam17ado87]

hi why is x3 a free variable in 3:26

[@gokcenafsin5366]

beautiful

[@nakulgupta182]

Amazing Explanation

[@informationisallabout8976]

Find the matrix P that diagonalizes A. Where A= 100120-352 . Are the eigen vectors linearly independent?

[@sumairafarht6809]

Just love it 😍

[@grantdrexler1210]

Thank you, that was incredibly helpful!

[@kahtanalobaidi1550]

I am gonna have this problem on my final. this is a great help

[@wildlife6612]

you are definitely better than my teacher... ♥♥♥♥♥♥

[@Emilie_in_june]

Omg my final is tomorrow and this literally saved me

[@vinayanpa126]

3:15 whats that. I didn't get it

[@sushmitha2537]

Hi, How did you get the x1, x2, x3 values? I did not quite get that.

[@sushmitha2537]

Trefor Bazett thank you so much. I’ll refer to your other videos too. :)

[@luckyleaf227]

How is x2 and x3 a free variable then? 3:32

[@luckyleaf227]

Trefor Bazett thanks! I thought it was leading 1s in each row. This clears things up

[@rajukotambari6197]

Nice video Prof. Please tell me technology you are using to make such videos

[@DrTrefor]

Just writing on my tablet with a greenscreen behind me

[@rajukotambari6197]

@@DrTrefor very nice.

[@rajukotambari6197]

What is the way to mask the background green screen.

[@sackofpotatoes1345]

My exam is in 2h. Wish me luck.

[@DrTrefor]

Haha didn’t get this until 3 hours latter but...uh....good luck!

[@sirjee3638]

Excellent teaching.... Thanks a lot Sir ji 👍

[@DrTrefor]

Most welcome

[@andrewson658]

can u explain more. like ?

[@yanzuo9740]

still can't understand the relationship between D and eigenvalues

[@STKeTcH]

why is x2 's'? Why is it not zero?

[@brianpeng3359]

Thank you teacher

[@themvlek]

Thnaks

[@ariturindamoses4707]

It's very interesting 🙏

[@kavya2807]

Thank u sir😎

[@RaggyTosh]

Edexcel FP3 2018 today in 8 hours

[@mhadiprabowo3430]

thanks you sir ,your explanation inspiration for me

[@DrTrefor]

So nice of you!

[@mahmoudal-ali5384]

Nice

[@LowieDM]

Give this man a Victoria Cross

[@mli3793]

You computed the eigenvalues wrong right? It is supposed to be det(λI - A) not det(A - λI)

[@yazanzoabi3268]

nice!

[@superchargedAMG]

don't you have to normalize your eigenvectors?