วีดีโอ

felt like a bunch of yapping at first but it all made sense with the final part

For dummies yeah right 💀💀

"Matrices are just linear operators in disguise", I like that line.

The explanation was top notch, but the presentation can use some touching up. Such as pacing, and cadence of your speech. Sounded like a high-school presentation by students who were strictly reading from a script they borrowed heavily from Wikipedia. The second way you could elevate the presentation is a better mic. But you don't gotta drop a ton of money to make a budget mic that gets rid of echo and room noise. Some eggshell bed foam and panty hose can be rigged around the mic and your voices will come through much clearer. Just make sure to speak facing directly into the mic.

Wonderful video

good vid ! but I think the animations could have been played a lot slower to give time to think (or pause and ponder as 3B1B says). leave more silent gaps and speak slower. thank youu

What do you mean by the adjoint being the "conjugate" transposition? I suppose conjugate doesn't mean the same as complex conjugate? Could someone please clarify

I’m not a math person and I never have been, I don’t understand any of the stuff in this video but I would love to. When I hear about quantum theory and all these important mathematical concepts I don’t understand them because of a lack of knowledge as well as a lack of comprehension in regards to the limited knowledge I already have. Where can I find out about the real world implications these things have? What do these things mean in regards to life?

So the spectral theorem is that all eigenvectors of a normal matrix are othagonal?

5:32 but you should also add in that sentence of what vectors the projections are FROM.

I'm just like Jackie in linear algebra; stiff hair and huge glasses.

small mistake in 0:44, the last matrix isn't real EDIT: ah, it's already in the description

What is the name of that font?

Well, it's a few years late, but I just stumbled on this video after watching th-cam.com/video/Mx75Kiqyaik/w-d-xo.html series, and this explains SO incredibly well!!! great job!! shame this didn't make bigger waves in SoME1 or 2, y'all deserve more credit!!

I love the style and every single word of this video!!

“Consider the following matrice with real entries” proceeds to put sqrt(-2)

I think that the video was really cool. However, it should be noted that many things in this video only hold when you’re working with finite dimensional vector spaces. But not all is lost: there is actually a generalization of this theorem in infinite dimensions. It is the spectral theorem for compact normal operators. Compact operators behave very similarly to operators in finite dimensional spaces. But a few more courses are needed to understand the topic in depth, notably courses in topology, complex analysis and functional analysis.

Many thanks for this video.

I wish all of this existed without the music. It really makes me want to not listen to the end, though It is a really good resource otherwise

opnesource the code from the video bro

This updated me ❤, im 21yo

The intro and the outro are hulariouussssss

Nice video. Minor nitpick: in your statement of the spectral theorem, you assert that "the eigenvectors {v_1,...,v_n} of T with eigenvalues {lambda_1,...,lambda_n} form an orthonormal basis for V" which is generally incorrect. The issue arises from saying "the eigenvectors" -- there are lots of them to choose from! If you write down a list of eigenvectors for all the eigenvalues, there is no guarantee that the eigenvectors are unit vectors, and (in the case of repeated eigenvalues) there is no guarantee that the vectors are orthogonal. For example, if T is the identity map from R^2 to R^2 (a normal operator), then you could write { (1,1), (2,2)} which would be a set of eigenvectors for the (repeated) eigenvalues {1,1}. However, this is not an orthonormal basis for R^2: the vectors aren't unit vectors and they aren't orthogonal. For normal transformations without repeated eigenvalues we never have trouble with orthogonality, but the unit vector issue can arise: for the diagonal matrix [[1,0],[0,0]] the basis B= {(1,0),(0,2)} certainly consists of eigenvectors for the complete set of eigenvalues {1,0}, but the second eigenvector is not a unit vector and hence B is not an orthonormal basis. To create the orthonormal basis of eigenvectors: first find a (typically non-orthonormal) basis B_k for each eigenspace of T by solving Tv=(lambda_k)v, then apply Gram-Schmidt to convert B_k into an orthonormal basis B_k'. (If you have repeated eigenvalues, you only find a single orthonormal basis for each eigenvalue, with number of vectors equal to the number of times that eigenvalue appears.) Then form the union of all these orthonormal bases to get the orthonormal basis B = B_1' u B_2' u ... B_n' for V. (The nice thing about normal operators is that all the different eigenspaces are orthogonal, so B will automatically be orthonormal.) A more precise statement: "If T:V->V is normal and {lambda_1,...,lambda_n} are the eigenvalues of T (possibly with repetition) then there exists an orthonormal basis {v_1,...,v_n} for V such that Tv_k = lambda_k v_k for k=1,...n."

Do you guys watch Sisyphus 55 your art style reminds me of his.

yoneda lemma ?

This was incredible! Need more!

wtf

Really good video, but please pause some between statements. Things move way too fast and it Is a bit annoying to always scroll back to pause manually.

W O W

Loved it! Great video, thanks!!

3blue1brown as sloppy as usual, always assuming a rushing graphic can make for a real explanation.

You guys are awesome, this video clears things up well

Things to know: 5 things I do not know and have never heard about Nice

I'm so glad 3b1b took this wonderful initiative. Your video, atleast for me, is one of the best outcomes which came out of it.

Will you tell me the font you used to wrote spectral theorem

can you put subtitles? the automatic captions are crap for mathematical terms.

Make more such videos

at 5:50 should not the eigen vectors be orthogonal? The picture doesn't quite capture their orthogonality.

The cuts were too jerky. Gotta leave equations and images on screen an extra half-second to second.

There's a nice proof of the spectral theorem projecting onto ever more nested subspaces that could have fit in here

Nice, thank you

The first ten seconds felt like what you feel when an advert on tv matches your situation so perfectly, you peek outside through the blinds and close them in fear.

I thought this was going to be about looking at the eigenvalues and eigenvectors you get from the adjacency matrix of a graph. They call that the spectrum of the graph too I think.

5:35 Exactly, lol. While I've probably learned this before, I think I learned different terms for everything making understanding much harder.

Does anyone know why adjoint operators are defined that way? About the applications, this theory is important to explore the realm of partial differential equations. The book by Olver on the topic is pretty good.

Excellent overview summary. Very interesting, informative and worthwhile video. I encourage you to make more videos.

Thank you for the review

4:01 Shouldn't that A matrix be A*T or A† (dagger) instead of just A*.

here for the comedy

at 3:50 I would have left-justified the equations, so that when you move the T the left term does not "change". It took me a while to check and double check to make sure nothing actually changed