Diagonalizing a Matrix

[@JordanEdmundsEECS]

You have no idea how incredibly helpful those short little pauses to backtrack a little and clear things up are. Thank you.

[@kronikevaporation6007]

Thank youtube

[@abdurrezzakefe5308]

Prof Strang is the best in Linear Algebra.

[@musikmakerfan]

MIT is lucky to have such a great lecturer.

[@dantemlima]

Gilbert Strang is a towering testimony to why superb teaching is much more important to learning than digital pyrotechnics. His conforting humble stuttering shows us that he still today is in awe by this formidable piece of mathematics and invites us to recognize and confront our own difficulties in learning. Thak you professor! I admire you from afar with great joy and personal enrichment.

[@jasonmarckx2912]

I"m in my first linear algebra course and am in awe of how immensely powerful this branch of mathematics is. MIT is fortunate to have a superb math teacher like Prof. Strang.

[@kunraiyan]

gotta appreciate how he said "I did that without preparing you for it", that was so humble.

[@tripp8833]

1.5x speed + Gilbert Strang = happiness

[@TheAllen501]

Much better than my prof who always tries to explain some very simple concepts in the most complicated fancy way so that it might make him look more qualified. The best prof should explain complicated concepts in the easiest and most comprehensible manner as possible

[@georgesadler7830]

These mathematical tools are very important in science and engineering. Dr. Strang is an incredible human being for linear algebra.

[@musicalwanderings7380]

Thank you MIT OCW! Prof. Strang is the ultimate contributor to education! Thank you!!

[@tusharadevi538]

I just love Prof.Dr.Strang's passion for teaching. He is such an amazing teacher. Having searched a lot of places to get an intuition about how different or same are eigen value decomposition and diagonalization of a matrix, voila, found all in one place. So glad to be learning concepts directly from a great mathematician like him.

[@maisamayhoon]

Tushara Devi again, Indians are everywhere 😀

[@bigboypal]

My Physics professors: *exhales in an annoyed fashion* "I really couldn't care less about the fact that I skipped 3 steps in my work while explaining a new concept this is extremely obvious and if you can't see it, I don't know how you made it into this class." Gilbert Strang: "I did a matrix multiplication I didn't prepare you for. I'm really sorry." Mr Strang I would literally die for you.

[@MsAlarman]

You are just a genius Gilbert! This is why you teach at MIT and wants to throw light on the shadows of ignorance in education round the globe. I am in bliss Sweet Angel!

[@alexandrefelicio3902]

THE BEST AND MOST PASSIONATE CLASSES I HAVE EVER WATCHED ON THE TOPIC

[@siddharthsingi6772]

This still remains to be the best video explaining this stuff!

[@MakerBen]

Thank heavens for this kind man :) More professors need to post high quality videos like this! This is super helpful! Thank you MIT!

[@aarifhussain3700]

A Life time asset ❤ priceless gift by The sir Gilbert Strang

[@meetghelani5222]

what an absolute joy of sitting through a course taught by prof. strang.

[@adityagaykar]

Prof Gilbert Strang, thank you for the explanation. I bow to you _/\_

[@c0t556]

Aditya Gaykar I’m on my knees

[@laurawitt8630]

I wish the professors at my university were this easy to understand!

[@ihbarddx]

I suspect it helps that the lectures are aimed at engineers, rather than at mathematicians. For whatever reason, they are certainly wonderful.

[@pubgplayer1720]

Amazingly succinct and powerful - so much important stuff in just 10 minutes. Thanks prof strang.

[@donotwantahandle1111]

Came here to learn why diagonalizing a Hamiltonian is important and learnt from a real teacher!

[@Raouli00]

I just had this in my lacture but didnt quite understand where the diagonal matrix came from but this cleared it up for me, thank you professor

[@malikialgeriankabyleswag4200]

So the column space of A or "transformed space by A" is the span of its eigenvectors! This makes sense of so many things you're the best Linear Algebra guy ever you legend

[@natalysalcedoguerra4078]

Thank you so much , excellent video.The best teacher that I ' ve seen until now.

[@Kneecap22]

Professor Gilbert Strang is the Stronkest at Linear Algebra! He is Lord King Captain General Warlord Supreme Commander of Linear Algebra!!!! Stronk!

[@ispeakfactslol]

literally THE BEST TEACHER...

[@rororoyourboat]

Thank goodness for videos like these.

[@yousafali7334]

First of all I would like to thank you sir for share your knowledge freely!I think it's wonderful for everyone who learn Multivariate analysis course....He/She must watch your videos.....Please share more of Calculus & other branch of mathematics...

[@shwang1579]

I can't believe that he can make this problem so easy for me to understand! Thx

[@martindahlgren7096]

You're a great lecturer! :)

[@leixia6415]

A^n = V * L^n * V^(-1) is actually eigenvalue decomposition of n-th power of A. Mr. Strang's illustration on how taking powers && taking differentials are like moving discretely && continuously are very a novel idea to me

[@KayYesYouTuber]

Dear Prof, You are a fantastic teacher. Thank you very much.

[@radicalpotato666]

More than 80 years old, but taught better than the faculty of most Math schools in the world.

[@tarlanahad]

Boss of Linear Algebra

[@fsiserir]

I let me go express my felling that you are the best Pr I have Seen.

[@jaamalarane959]

we appriciate MIT and youtube for giving us our brain food thanks proff gilbert strang we also have herb gross for calculus

[@alexanderwhittemore1491]

"That's very nice... that's very nice..."

[@teutadomi5823]

You are the best Prof Strang!Thank you!

[@umehmoses8118]

I just love you Professor.

[@Froodomir14]

a very good teacher.

[@OriginalSuschi]

There is also the notion of simultaneous diagonalization, meaning two diagonalizable matrices A and B consist of a basis of vectors which are both eigenvectors of A and B at the same time. Given diagonalizable matrices A and B, the subset of all diagonalizable matrices C which are simult. diag.able with A and B with the same base change matrix, they actually form a subspace of Mat_nxn(K) (the vector space of nxn square matrices over the field K)! And since A and B are obviously simultaneously diagonalizable with themselves, we know (for A=/=0 or B =/=0 matrix) that this subspace is not just the zero subspace. Furthermore, multiplying two matrices which are simultaneously diagonalizable yields a matrix which is again diagable with the same eigenvectors as basis of vector space, and the eigenvalues are just λ1μ1, λ2μ2, …, λ_n*μ_n. And also adding them keeps them simult. diagable. One can also show commutativity under matrix addition and multiplication, anf left and right distributivity is given. Right now these form a commutative ring (since for every C, also -C is inside, 0 and 1 are also inside and unique). If we now let A and B be invertible, all simultaneously diagonalizable matrices with A and B are also invertible (except 0). Since now every matrix in this subset except the zero matrix has a multiplicative inverse, we get a new field! This field is embedded in the field of all invertible matrices which commute with A and B(but I don‘t know if these are the same or not)

[@riteshvesalapu8373]

I just love the lectures. You are the best sir. Kudos to you.

[@saiveeryamahadevan1266]

Prof. Strang is AWESOME

[@LibertyAzad]

And this particular video was exceptionally helpful to me. Thank you!

[@vikashdhanabal8984]

the best maths teacher in the universe including the ultragenius aliens in the space

[@Zephyr-tg9hu]

Reviewing for my final. Thank you so much for making it so easy.

[@hujiawei6774]

he makes linear algebra so beautiful to me

[@infiniteprogramming6214]

This vid has made my life!

[@HassanOmariprofile]

Thank you Dr. Strang, great video indeed

[@sanketgandhi3139]

Does V inverse always exist?

[@oudarjyasensarma4199]

ok I'm a little confused around 5:03 if we multiply both sides by v(inverse ) how are we getting v(inverse).A.V = "capital lambda(evm)" we're supposed to get A = evm right?

[@infiniteprogramming6214]

We can't equalize v*lambda*vinv with v*vinv*lambda as matrix multiplication is not commutative

[@flyingbirds6794]

that is a really absolutely wonderful video!!Thank you very much

[@DJ-yj1vg]

This guy is incredible

[@AnkurDeka]

You are such a wonderful teacher!

[@chrischris948]

If only all professors were half as good as Professor Strang.

[@sushantpoudel4372]

4:20 How can V have inverse? Isn't it a non square matrix?

[@emenikeanigbogu9368]

11:19 Professor Strang gave us the secret to time travel

[@deepanshusharma8226]

If time travel was possible where are our guests from the future

[@dalisabe62]

This of course works only if V is a square matrix and non-singular; otherwise, inverse V does not exist and the entire technique crashes. On the other hand, the SVD decomposition works for all matrices even those that are singular, because the method incorporates the transpose in place of the inverse.

[@LuisGarcia-te5cr]

Thank you, very helpful explanation.

[@SL-cr3vh]

Understood very clearly, thank you very much! :)

[@noobody138]

What a great teacher!

[@sgzhou6242]

What a great mathematician!

[@Genshin_suit]

only the rocking star of linear algebra can do this

[@vinaykumardaivajna5260]

Super helpful and thank you so much

[@HS-zu3tu]

Salute to you from Japan

[@albertyeung5787]

simply great

[@dimasghiffari9441]

makasih eyang strang :) jadi enak dan simple kalo bapak yang ngajar

[@GigasnailGaming]

love this prof.

[@225discovery]

such a great explanations.

[@deveshvaish8182]

So for any N X N matrix do we always have N eigenvalues and eigenvectors?

[@sftekin4040]

This is beautiful...

[@emaniratnavalli3208]

Thank you so so much sir.

[@User-su3ut]

This video makes me wish KZbin had a superlike! 😅

[@KenRubenstein]

Gilbert is a good guy.....

[@bismeetsingh352]

This is beautiful!

[@RobelDelelegn-y9t]

Thank you.

[@marcodong3749]

Oh damn, You enlightened me. Thank you very much!

[@gogolgullu]

thanks gil

[@cianorourke7195]

Thank you

[@andrewl5267]

Do you get the same eigenvector multiple times when an eigenvalue has an algebraic multiplicity greater than one?

[@AbhishekJha-sz7cp]

both strang and mathematics are really cute

[@safatkhan6839]

04:20 Can someone explain to me how multiplying with V-inverse would be valid? V isn't a square matrix. Doesn't that mean that it cannot have an inverse? In other words, V-inverse doesn't exist!

[@nadeemnazir295]

SAFAT KHAN dear,it is always square because here r two vectors in V that doesn't mean that itz not square...and V would b always invertible because V contains linearly independent eigen vectors X1,X2,...Xn.

[@safatkhan6839]

I think I assumed that the entries in V were individual scalar quantities. If the two vectors in V each have 2 elements in them, that would turn V into a 2x2 matrix and as such a square matrix. I don't see the same happening for vectors with more/less than 2 elements in them. Is my understanding correct?

[@maartenmtr3022]

great lecture thank you

[@B.Shouvik17]

he is a legend..... till 18-03-21 I was remembered that formula.......... GOD real GOD

[@VFORVICTORY1000]

legend ,most of the tutorial didnt say the whole thing ,they just use the definition.

[@swagatodey4912]

Thank you thank you

[@subhramondal134]

cleared a lot of doubt❤

[@prateeksahni2923]

beautiful

[@yanningli8362]

OH so clear!! Thanks a lot!

[@aashsyed1277]

he writes z just like my parents! shout out from really far away thankkkkkkks

[@bradypeng621]

Thank you sir

[@Jinouga502]

"Eye"-gen vectors and "eye"-gen values.

[@innocuousobservnt2325]

thanks

[@gregarityNow]

GOATbert Strang

[@adityanatu]

Each time you operate the same matrix on an eigenvector, you get back the same vector, just multiplied by its eigen value. So it's rather obvious that any n-th power of any matrix will have the same Eigen vectors, and Eigen values just get raised to the n-th power!

[@dariodrigo3778]

Why was V invertible?

[@kencen4099]

It is so helpful.

[@yuntongzhou1185]

"now that I have it in a matrix form here I can mess around with it." lol in lib