❖ Finding Eigenvalues and Eigenvectors : 2 x 2 Matrix Example ❖

Part 1 • ❖ Introduction to Eige...
This is the second video on Eigenvalues and EigenVectors.
Finding Eigenvalues and Eigenvectors : 2 x 2 Matrix Example.
In this video I outline the general procedure for finding eigenvalues and eigenvectors for an n x n matrix and work an example using a 2 x 2 matrix.

[@patrickjmt]

Hi all! Please post comments, questions and anything else on your mind in the comment section! While I try to answer as many questions as I can, there are more questions than I can answer by myself so if you are someone who might be able to help, please do so! The community is so much better when everyone is involved and helping each other out. Our strength is our willingness to help each other! Also please feel free to like or dislike the video. Your feedback is always appreciated so that hopefully I can make my future videos the best possible! Thanks and happy studies!

[@deckeresq]

Please definitely post a 3x3 example! This is super helpful.

[@zeeshanchaudhry3324]

Can you generalize discuss the implications of the cases where the eigenvalues come out as imaginary?

[@orange399]

How did you found adjoint of 2x2 matrix youre moving very fast at 8:20 finding B matrix? Because its equal to zero matrix please reply fast i have exam two days later

[@apollosleaf731]

Do we have to follow all these steps or is there a faster way to go about it?

[@bhumikasrineel6799]

Please take an example of solving 3×3 matrix too

[@visionary6498]

Do you ever think about how many people you've helped become engineers or scientist? You are really doing humanity a great cause my friend, thank you!

[@patrickjmt]

+The Visionary I'm sure I've helped a few here and there :)

[@bigbubbadave7370]

you are the only reason I am able to pass calculus and linear algebra. My teachers are great, but for some reason this "one on one" examples are really what gets me to understand it.

[@Ghost572]

Its because he's too the point and writes things in steps, like i was watching some other videos of people explaining this and there just waffling on and jumping between things without explaining them as if the person has done them before. There isn't really much excuse when your doing it by a video because you can prepare the most key and important points and how to say it, before doing the video.

[@andrebayan]

Every time I have written a Linear Algebra and Calculus I & II exams/tests, I always get a, "What did patrickJMT teach me?" moment. I'm probably going to have those brilliant moments for the next 4 years... Then I will come back to this same video and thank you for catapulting me into graduation day with a degree in Mechanical Engineering.

[@balkdermins1528]

hey where are you studying Mechanical Engineering > .. Im studying it At the University of Guyana

[@AldosWorldTV]

I have litterally skipped 80% of my linear algebra class this semester and have over an 85% solely based on studying from your videos.. I love you man

[@KH-ft4ut]

Just did the same with an exam coming up in 2 weeks. Why do we do these things to our self?!?

[@chukwumannadi706]

LMAO!.

[@Sam-wt1cx]

@@KH-ft4ut Exactly.. My exam is in just after 2 weeks and I ve started to look at things just now

[@KH-ft4ut]

@@Sam-wt1cx Good luck mate, I passed my exam. But I still have no idea what Eigenvalues or eigenvectors are actually useful for.

[@jgc9199]

The Duke lol I guess I’m lucky my professor is not bad although these videos help me out cuz I’m taking two math classes and I get confused at times.

[@Poodz_]

It's 2am, I have a test at 9am. There will be literally 2 questions on a 2x2 matrix. 1. Find the eigenvalues of this matrix 2. Find the eigenvectors of this matrix. This video wins the internet.

[@adamc5478]

+Allegro Crab introductory ?Linear algebra?

[@Macca-95]

+Allegro Crab It's currently 1:15am here, my exam is 9am. I can relate :P

[@Poodz_]

HaydenMcM1 you're australian too? good luck m8.

[@Macca-95]

***** Yep, thanks mate :)

[@edacordon126]

its already 2016 but I can relate to what you said. its 2am and I have my exams later at 9am. 😂😂

[@ReWolWerKillZ]

fools cant even explain this in an hour. patrickjmt is a beast.

[@fairax]

When they explain this stuff in an hour they just rush through it and don't explain anything, Patrick is a life saver

[@ReWolWerKillZ]

Eric Da Silva Uni profs are usless bro

[@ProductionBandit]

i guess thats why there's that saying "those who can, do; those who can't, teach"

[@ghostf6321]

Yea seriously if it weren't for patrickjmt and other KZbiners I would have no idea wtf is going on in my diff eq class.

[@patrickmoloney672]

University professors are not there to teach you. They are there to provide you with material that you should already know. University isn't high school.

[@MrAngeloungery]

I like how you do math, you know how to explain it in steps, I wish my instructor can work in step, thank you so much patrick, God will reward you my friend.

[@patrickjmt]

i am already rewarded by feeling appreciated in this world as your comment does for me ;)

[@सत्यमेवजयतेन्यूज-ल1ट]

Angelo Ungery the best regards and best wishes

[@MFDan69]

let me just say... you are single handedly saving me this semester.

[@jianeecarrasco9106]

Literally !

[@jamesarokiy94]

You sir, are a God amongst men. Thank you for these videos, you legend.

[@patrickjmt]

+Flambe nah, just a normal guy trying to help in the best way i know how!

[@kentrelljackson]

+patrickJMT JOHN 8:32 :) #lifeabove

[@reiantaank3314]

+Flambe I don't think it's right comparing him to god. I mean he's good... but he's no Patrick.

[@superchargedAMG]

definitely a god among men

[@rgseven6557]

Dear Sir, I sincerely thank you for taking the effort to create such useful videos. I was reading through my notes and textbook till i began to panic. I have an exam in 2 days time and I would no have understood this topic if it were not for your kind help. Best regards from Singapore.

[@brendanreidy4705]

You helped me in 12th grade with AP calculus and now you're helping me as a senior in college. Thank you my friend

[@ObviouslyyWrong]

I will never understand why so many professors are unable to explain what a theorem states and then go on to give a simple step by step answer like this

[@hxqing]

也许，这些理论在某些教授的理解能力中，还不是十分清晰的。思路未清，词即难达。 Maybe,some professors have not get the clear understanding in their mind. No one can explain more clearly than he think.

[@sdlion7287]

I think that since they have done it so many times, and have realized the importance of the _concept_, they focus so much on the theorem and not on the operation. It is important, but a lots of times you understand the importance and _can use it_ after you are familiar with SEVERAL tools. One day, they just click and understand them better. That's not the situation that students are in, they ask us to remember lots of things that stack upon each other to build new things and they tell us "this (ALL) is important"... well, we (still) not care, we don't have the tools to care either.

[@generalsweets4542]

I know this video is 9 years old but even to this day you are helping others including me to achieve their goals in college and actually aid them in understanding fascinating mathematical topics. Thank you and I wish you well :)

[@malu8710]

I've been trying so hard to understand this! I really understand math when someone explains it step by step to me. I was stuck after getting the eigen values and never knew all you have to do was assume a value for x1 or x2. You helped me out a lot!

[@Chino747747]

doing maths along listening to your videos is blissful!

[@patrickjmt]

Ha, glad you feel that way!

[@heldermachaieie1706]

Thanks for the video, sorry, can You please explain how you came to 3R1+R2 >R2 when determining the eigenvectors? Thank you in advance

[@MCabossart]

All of your videos are very helpful, and I recommend your channel to all my fellow engineering student friends. Thank you for creating all of your videos.

[@stanleychan2099]

I read so many ****ing tutorials online explaining this concept, none of them came as clearly as yours. Thank you for contributing knowledge to the public.

[@stanleychan2099]

May I ask for an application example for Eigenvectors and Eigenvalues such as Page Rank algorithm?

[@amizan8653]

I cannot express how grateful I am for linear algebra libraries, saving me from having to do any of this

[@estebanlopez1701]

Jajaja I couldn't agree more. It's pitiful how many teachers are reluctant to integrating technology to subjects like this

[@fredmichael8548]

i must admit that left handed people are always somehow gifted and special. thanks for helping man !!

[@HaydenHatTrick]

In that case, my instructor is a masochist

[@TheZwhyX]

+HaydenHatTrick same

[@cjlooklin1914]

Lol, I think you mean sadist, my Eigen value were 5.372 and -.372.......

[@SuperLuckyKill]

+Cj Looklin Tell your professor to screw off. I feel so sorry for you

[@Recon13x]

lol i had a three decimal eigenvalues. and had to do inverse off a 5x5. not terrible but he wanted adj(5x5)/det(5x5). This means that ill end up having to find the det of 25 4x4 matricies

[@datascientist3663]

Oh noooooooooooooooooooooooooooo

[@sanjarkarshiev5394]

March 2019 and it is still very useful video, I appreciate your teaching style!

[@dannygabor6154]

Thank you so much for taking your own time and energy into making these videos. You have helped understand so much more then my teacher and a lot faster. I just wanted to say thank for all the hard work you put in to help people like me out. Your awesome!

[@teddy5474]

I was bad at math since I was in junior high school. During my first semester in university, I found this guy's videos and watched them to help myself get over the calculus classes. Today I am taking my last undergrad final exam, and I am still watching his videos. And perhaps I will come back to watch the videos after two months when the graduate school begins

[@mjackstewart]

How did you know the “plug in values” to create the Eigenvectors? Would any number for x1 do if you multiply it by the coefficient of x2?

[@plpI1]

yes any number you can since it just a scalar. For simplicity just pick number 1.

[@captaincosh2703]

got a maths resit tomorrow and this video has saved my life, the lecturer that teaches me maths made this so much more complicated then it needs to be, so thank you!!

[@abdelrahmansaid844]

Is there a word more than thank you.ahh I appreciate this. no there must be something better. You are a complete University.

[@lindsey5095]

Thank you so much for breaking this process down in such detail. It really helped me understand what was going on for more complex and high n matrices in the course I'm taking.

[@janeliu6211]

Thank you for this clear and helpful refresher! If you ever have the time, it would be great to see some videos explaining related applications of linear algebra in data science, like PCA and SVD. It’s hard to find clear, succinct explanations for them!

[@Wake_Up_Its_The_NWO]

You have literally helped me 4.0 all 3 calc classes, my proofs class, and now linear algebra. Thank you!

[@hamxa_says]

my teacher teaches a same question by very complicated method but your method is so understandable thnks sir :-)

[@winnervictor4401]

I never understood this when my lecturer taught me.. but after watching your video, I got a clearer view and understanding about the course.. thanks for the explanation 🙌

[@keithgsa1]

I wish my physics lecturers at university had been like you.

[@attahsalifu1091]

Prior to opening this video, I'd spent the past 2 hours fruitlessly trying to understand this topic using textbooks. 10 minutes here and I understand already, thank you.

[@turnawayandgo]

You are the best at explaining! I love your videos

[@patrickjmt]

Thanks!

[@XXgamemaster]

+patrickJMT Hey Patrick. For the first problem with eigenvalue T = 8, you used a row reduction method for finding the vector x. In doing so, you made the second row of the matrix have all zeros...BUT for the second eigenvalue T = -2, you made the first row have all zeros. Can you explain why you used that method and didn't use the same row in both cases?

[@Fabizy]

+JC Tutoring it's the same if you make the second row of the matrix have all zeros, you still have same result

[@syd_jzx100]

For the eigenvalue -2, I divided the top row by 3 so instead of 9 and 3 on the top, I ended up with 3 and 1. From there you can do Row1-Row2 and get (3,1) on top and (0,0) on bottom row.

[@iraeshkumarghosh2669]

I'll say it once: BEST VIDEO on this Topic!!!

[@javadhikmati4941]

Can you please put all linear algebra videos in one playlist ? that would be wonderfull but thank you Patrick!

[@MrHerbie24]

We all just need to subscribe to your future youtube sessions rather than wasting money and time only to feel totally confused and sleepy in University class learning about these...Thanks Patrick. You are a legend!

[@ccashman5]

Great vid, but why do you just choose any number to sub in, wouldn't you get a different value if you subbed 2 instead of 1?

[@aditya-ex8hh]

2 minutes in and im already understanding more from this than my slides from my uni website, god bless you

[@Grantttteeee]

When choosing a value for x1 for the eigenvectors, does it specifically need to be 1? or any value? and how do you know what value to choose?

[@chockesondicks7505]

Yeah, like you can also use the letter A for example X2= A and X1=3A

[@seansethi196]

I have the exact same question, choosing 1 seemed arbitrary but I do not understand why it was chosen.

[@moatef1886]

@@seansethi196 It ultimately doesn't matter, because you just need one any one eigenvector. Remember, any non-zero scalar multiple of an eigenvector is also an eigenvector for a particular eigenvalue.

[@seansethi196]

@@moatef1886 appreciate the reply, thank you!

[@efosaomoregie5246]

I'm from nigeria and I have exams the questions they set are different from what they teach us and the man has made me understand what I'm learning

[@germanasaroti]

greatest youtuber!

[@C4rb0neum]

I've seen two video's about eigenvalues and eigenvectors and am amazed. Amazingly clear instructions.

[@Khalil9497]

Why we let x2 = 1 I don't understand, please help

[@bobbutts524]

It doesn't matter what you set x2= to. you can pick any number you want because any non zero multiple of eigenvector xbar is an eigenvector

[@bobbutts524]

I don't like how i explained that... let me do it again. Once you find the relation between x2 and x1 it doesnt matter what numbers you chose. You may pick any x1 to get a corresponding x2.

[@Khalil9497]

ah okay, thank you so much I've thought about It but I was uncertain, thank you so much for your answers, and sorry If I make english misspellings, I'm moroccan and we study In french

[@andyroutine132]

bob butts Love the explanation. You had it all messed up at first but you came back with the second post and cleared it up. You could have left it all messed up and probably nobody would have noticed but you came back... You came back. God damn you got a bright future. Best of luck in future endeavors and shout out to any who enjoys linear algebra

[@skylarkenneth2407]

​@@aussiecomrade5972 There is a subspace of eigenvectors with x1 and x2 corresponding (or relational) entries. (orignal matrix) * (subspace of eigen vectors that you choose from)= (eigenvalue) * (eigenvector chosen).

[@matthewknott9510]

You sir, are a lifesaver.. Finally taught in a way that I can understand fully. Great job man, and thanks.

[@taninhanutry]

Do you have any video on diagonalization of matrices ? Thanks for the helg witn Eigenvalues-vectors :)

[@aman7697]

I love you baby 😊😍😘😚😗😙

[@alvianfahru2473]

i'm doing research on Principal Component Analysis, and your guide is so useful. thanks .. it's very easy to understand.

[@t-petkovpetkov7353]

in 7:10 you wrote 3R1+R2→R2 How do you pick exact these values ?? for row one and two

[@matf6661]

Sorry for the late reply but it may help others: You pick numbers that reduce the bottom row to zero, here 3*R1 is the same as -R2 so by adding them together they cancel out, giving the desired zero on the bottom row.

[@gamersden30]

Thank you from Australia

[@HariPrabhuJAP]

Welcome from india

[@joelrodriguez1232]

Wow I read lots of books on this topic and now I understand it. You are the best!

[@missghani8646]

your videos are simple and understandable thank you. just a question how did you pick x2=1? is it because 1 and 3 are within the matrix or can we chose any number?

[@shirashira8871]

yes i want to know about this too

[@sauceandalouse2624]

@@shirashira8871 I'm in the same position hahah I want to know too

[@sauceandalouse2624]

@@shirashira8871 got the answer ! you can see in others comments : "ultimately doesn't matter, because you just need one any one eigenvector. Remember, any non-zero scalar multiple of an eigenvector is also an eigenvector for a particular eigenvalue."

[@shirashira8871]

@@sauceandalouse2624 Thank you for your explanation:)

[@forerunner6228]

You make all lessons simple to understand 😎

[@Michb3ck]

I can pinpoint the exact moment when you lost me :D 7:00

[@peterlee5726]

Hey it's been 3 years but yeah.... WTF is row reduction?

[@ducksoop.x]

@@peterlee5726 In typical University Linear Algebra classes, we have to reduce our matrices manually. This can easily be done in most calculators or something like MATLAB.

[@Josh-zk7pf]

Peter Lee I know your comment is old, but KZbin for his video called “gauss-Jordan elimination” that’s what row reduction is

[@NayAPakistanJobs]

but why we need row reduction here

[@BoburMamadzhanov]

Could anybody explain it to me please?

[@tarakenabay6384]

In addition to the world class explanations, I appreciate the encouraging statements at the end of your videos. It really helps me to keep going.

[@Tedlasman]

That sharpie though. So squeaky. The video is fantastic though.

[@MrJaros]

Kind of a comforting sound tbh

[@fengpan1420]

you teach better than my Prof, you teach better than the author who wrote the textbook. Your videos give me hope to ace my final exam! Thank you so much sir

[@victorserras]

“Unless the instructor is a real masochist” thanks for accurately describing my math teachers

[@chillvibing8374]

i love the way you teach, I've been struggling in my Matrix Algebra course and your videos help me tremendously, thank you

[@NakasDougen]

Doesnt that mean there are infinite eigenvectors?

[@staceyburchette3276]

Yes, once you find an eigenvector, any scalar multiple of that vector will also be an eigenvector.

[@booceyboo446]

9 years later and still helpful. Thank you :)

[@MidnightKnight16]

You think you can do a 3x3 matrix?

[@rwh777]

I'm assuming ur stuck after you've written down det(A-lambda times I). Go back to your notes on calculating the determinant of a 3x3 matrix. It's just a bunch of tedious calculations at that point, and you end up with a cubic equation which of course is a little more difficult to factor to determine the 3 roots which then give you 3 eigenvalues. Then its just a matter of substitution as Patrick showed you to get the eigenvectors.

[@matildabjorkman6713]

You explain everything so well, I finally understand how to find the eigenvectors!! THANK YOU!

[@AceofDlamonds]

Finding the Eigenvector from 6:00 to 8:25 is a bit confusing, namely the row reduction and seemingly arbitrary picking of values for x1 and x2. I tried to do it step by step with another problem. I get lost there. Can't find the Eigenvectors :(

[@supahotfire8886]

Instead of thinking of the problem as: Find THE eigenvector. Reword it to: Find AN eigenvector. When he gets the matrix into Reduced Row Echelon Form, - research this to understand why we row reduce - any real number we plug into the equation he writes at 7:43 for x1 or x2(only pick a number for 1 value and solve for the other) can reduce to {3 1} like he got, so that's why its an arbitrary selection. Try it!

[@AceofDlamonds]

@@supahotfire8886 Thanks. :) I'm out of school now but Ill try it lol. I think i did accomplish this in one of my later courses.

[@litjew6669]

are you an engineer now lol@@AceofDlamonds

[@AceofDlamonds]

@@litjew6669 Hehe...i'll just say I wish I could rewind time and study computer science instead 😂

[@ahmedelfarra7674]

I've never commented on a KZbin Video before. But this right here is just amazing. Thank you. My professors make it feel impossible. Thank you so much. I'll open all the videos you made, and see the ads. You deserve it.

[@azar9383]

7:10 where did the 3 come from? 3xR1 + R2 = R2 ??? !!!

[@samanthaseetaram1516]

Can someone answer this plz????

[@yoga5631]

It's a variable, i think he is using the gauss Jordan formula

[@hpyakurel]

he is trying to reduce it in echelon form.

[@conradpintoes]

You're a good soul! I have benefited from many of your videos. Thanks for the help!

[@Jimjamatic]

As soon as you get to the row reduction, I'm lost. Dammit....

[@Daveuuuh]

Without doubt the best video on the web for eigenvectors and eigenvalues. Thanks a lot this masterpiece.

[@tp7886]

My instructor must be a masochist. Had a 4x4 on the last exam.

[@nefo706]

You are the main reason why I'm passing my Math and calculus courses...Respect Math Boss

[@CrankingAllDay]

Have to do a 4x4 by hand.... fml

[@goodhore]

the row reductions of these examples were always so charitable, the test, meanwhile, was designed to split your head.

[@nickycat101]

my instructor is a masochist...

[@MrJaros]

more likely a sadist...masochism is self-inflicted pain

[@floydnelson92]

I was scrolling just to find a comment like this, lol

[@CMDParodies]

You actually teach properly, I did all this in my A-levels, and going into 2nd year of uni I'm brushing up on it. You go into the right amount of detail necessary and don't take what I'm predicting to be 3 lectures (150mins) to get there :P thanks

[@bbrealey]

Thank you! After 1 viewing of this and working out the example on paper, the concepts are so clear now.

[@meensbeans_]

Your explanation helped me tons! My textbook couldn’t even assist me, let alone my lecturer. Thank you 🙏

[@otooo_0500]

めっちゃ、わかりやすい！！ これで、テストも頑張れそうです！！！

[@rezaerfan7096]

Wow! Nice! Thanks! I read the slides of our class 3 times and understand nothing! This vid was a great help thank you!

[@dp0813]

I echo the comments below; great videos on this topic! I took LA 3 years ago and I didn't realize how much review I would get on determinants, row reduction, etc. Thanks!!

[@MrMekNoodle]

Great video, doing this in matlab and it's just great to know what the maths behind it is rather than just putting numbers in and getting numbers back, thanks a lot!

[@ChefFarisMom3]

Thank you sooo much for teaching in a wonderful way and turning complicated things into easy things...👍👍👍

[@MichaelBrown-fl9lc]

This helped me understand what "diagonalizing" a 2x2 matrix for the schroedinger equation actually meant in my Solid State Physics class so THANK YOU!!!

[@patrickjmt]

happy to help :)

[@ksoflyineer]

I sometimes cut class to watch these videos, soo worth it!

[@josephpearce2201]

this has been very helpful, i was trying to watch a khan academy video but had lost me with it. this video really cleard things up for me :)

[@benjaminb320]

1 hr before the exam and I'm not worried! Thanks Patrick

[@marcobrasilgv]

Amazing, I learn here in 5 minutes what didn't learn with my teacher in 2 hours. Keep up the good work Patrick.

[@mohamedelgogary4764]

Patrick, You are a life saver. Thank you Very Much for the Simplicity and the Outstanding Work.

[@a.darwish469]

I just wanna say thank you for all the great help .. you are really doing something useful for all the students . I wish someday to do something as good

[@md.shahinurrahman747]

Thank you Mr. Patrick. in the classroom I can't understand then, That's why I had to suffer. Now I'm clear. Thank you again.

[@amandasirisoma9853]

thank you so much! You explained the concepts in an easy-to-understand way, and I actually understand them now :)

[@Momen-]

You're better than the doctor who teach us . Thank you so much

[@goldmanmwasangoko9711]

Patrick you are a good tutor such that your naked all concepts to my sight. keep it up. Thanks

[@ylvaeriksson5761]

I´m so greatful for your explanations! If it weren´t for them I could never fix this course.

[@paokalexthes]

simple and very exaplanatory at the same time, thank you so much

[@kode24762]

learnt a lot of great technics from this video that can be a little bit hard to grasp from a textbook...

[@michaeltheisen]

Thank you for providing such a good example AND showing a good method to check. This helped a lot, thank you!

[@bettybanda1036]

Wow thank you so much❤ I won't forget this as I write my exams today 😊

[@JackRule16]

This is a great video and I don't want people to think I'm bashing it, but in a general sense rather than simply picking an X1 or X2 and finding a specific eigenvector my professor often wants us to write this in a general form. In this example it would be something like (3X2 for entry 1 and X2 for entry 2 in the vector)

[@BenBush96]

Thanks for spending the time to make these videos. They're make things a little more fun, but more importantly easier to understand and digest.

[@ciarabyrne1931]

This is literally getting me into my next year of college,ut thank you

[@NigelBait345]

Thanks a lot dude, your videos are helpful to boost up some dry areas......

[@ayecob558]

U deserve a nobel peace prize for your contributions to society fr