Infact, proving it for diagonalizable matrices is enough. This is because the determinant is a polynomial (and hence continuous) and trace is a continuous functional, and that diagonalizable matrices are dense in the space of matrices

Awesome... thanks for sharing. When I took ODE, my professor introduced me to matrix exponential and I was blown away. Sometimes I wish I could go back to that time.

Yesss a live stream on Lie algebras, I'm already hyped for that!!!

13:24 Element (1,2) should be -e^3+e^6, not e^3-e^6 - signs are swapped, but corrected when the board is erased and updated a few seconds later.

The corresponding operation to the Lie group multiplication in the Lie algebra is not the Lie bracket but the addition. Indeed, we have exp(x) · exp(y) = exp(x + y) IF they commute (that is [x,y] = xy-yx = 0) .

I thought i knew a good amount about math when i finished calc 1 senior year. I was a newborn then. Math is so beautiful and interconnected. Thank you for sharing this with me. Ill be excited to watch the livestream :)

This was a good refresher about diagonalization. Thank you.

To find the determinant, you don’t need to expand the multiplication: det(P M P⁻¹) = det(P) det(M) det(P⁻¹) = = det(P) det(M) det(P)⁻¹ = det(M) for all matrices M and all invertible matrices P. Edit: Maybe I shouldn’t comment before finishing the video. Now I notice that you used this exact trick for the general part. Still, it would’ve saved some effort on the example too.

super interesting! great content as always, thanks michael

It's been eons since I've diagonalized a matrix, so nice reminder! Looking forward to the stream, I learned some basics of group theory but nothing nearly as advances as Lie algebras.

That last part about Lie groups and Lie algebras was really illuminating. I can't wait for the livestream!

Hey Dr. Penn. Thank you so much for these lessons. I've been out of college as an electrical engineer for about 2 years now. When I took linear algebra, I had some personal issues that led to me absorbing little from that class, barely passing. This video is such a good refresher on some of those concepts I've forgotten, like how to find null space.

Wow. Thank you Dr. Penn. And thank you again for your excellent differential forms playlist.

Wow. I had no idea matrix exponentiation maps from a Lie algebra to the Lie group. Very cool. I need to know what the Lie bracket does!

I took Linear Algebra 5 years ago, I didn't study any further math (not at least for algebra) and I could follow you through the entire video, it is not only a beautiful property and a great explanation, but also brought me nice memories. The connection to Lie groups and algebras got me intrigued, I will read about it! Thanks for the amazing work, keep it up!

I loved the tie into Lie stuff! Thank you for posting.

This has given me so much insight, thank you!

Wow, this is a very, very good video. I was familiar with determinant calculations but that's all I remembered, and it was very clear what was happening all the way through, and made Lie algebra seem like an intelligible concept too. High level math content!

we had to prove this when I was in school... one of the most satisfying math assignments I ever did. Thanks for this video!

This is one of the coolest videos I've seen in YT about mathematics. Wow.

SUPER cool. By far my favorite math channel.

Incredible video! Thanks a lot for this! @12:53 you made a small mistake in the the upper right entry (a factor of -1) This happens to me all the time when I teach. Glad to see you're human too and glad of course that it was corrected on the next blackboard.

This is an awesome video. I didn’t learn much about Lie group and Lie Algebra in grad school. Looking forward to your next video, or perhaps a series in the future.

Can't wait for the Lie algebra stream. Sounds really cool!

Very pleasant to be able to follow the discussion at 3 different levels, one after another.

I know little to nothing about this branch of Math, indeed your explanations are so much well done that I was able to follow everything and .. this video is beautiful, it merges tons of knowledge in an armonious way. Lovely Thanks for sharing!

Lie algebra livestream sounds amazing, can’t wait!

Such an excellent problem and presentation. Your Linear Algebra students are fortunate.

Just Great ... Please more of these (Matrices and ...) Thank you so much Professor

Hi Prof Penn..thanks for another wonderful exposition..looking forward to the Lie Algebra live stream...

AWESOME!!! I've been waiting for more Lie algebra content, thanks for posting this!!! Can't wait for the livestream!!

It's for proffesors like you that we students remain inspired. Appreciate your effort a lot!

Wow you have my attention, Sir - thank you

Great video, this is gonna take me some time to process.

Yeah I'm definitely interested in the livestream, what day is it?

very cool video! thank you!

25:44

Ooh the big picture! @Michael Penn Have you done any video about Fourier Analysis and FFT? If not, do you have any plans for doing so? Cheers, nice job! 🖖🤓

Lie groups and linearalgebra nostalgic memories, lots of fun

Linear algebra will always be a favorite of mine. Unfortunately, I haven’t had to use the stuff

Loved this!!!

Great teacher !

Would love to see such livestream about Lie algebras

Hi Michael I love watching your videos - have you ever thought about attempting STEP problems which are similar style to IMO but for admissions to Cambridge here in the uk

More videos like this one! I love Lie algebras

Your content is awesome.

Awesome, thanks!

Best video yet!

Great video !

Fantastic! This really ties together so many loose ends for me in a concise and clear-headed fashion. Definitely looking forward to the livestream!

We use that on Quantum mechanics.

Amazing subjects. After 10 years of so many maths, and still I've got so much to learn... When are you planning to do your Lie Algebra class? Tell us through an Instagram story!!

It will be awesome to see a proof of Baker-Campbell-Hausdorff formula. Thank you for this professional and good lecture.

There is really beautiful proof of tr(A) = , it goes like follows: = Now let's recall, that the sum of roots of polynomial of n'th power, written in form x^n + a1 * x^(n - 1) + ... equals to -a1 Thus we need to find the first (or the second, depends on your notation) coefficient of the characteristic polynomial of A, or the polynomial det(A - xI) Now we just need to substitute determinant by the definition through permutations, and notice, that only one summand contributes to the coefficient in front of x^(n - 1), and that is the product of diagonal elements, i.e. (x - a_00)(x - a_11)*...*(x - a_nn), and from here it's not hard to see, that the coefficient in front of x^(n - 1) in characteristic polynomial of A equals to -tr(A), so the sum of eigenvalues of A equals to tr(A) P.S. Sorry for mistakes, I'm not really experienced in explaining math in English

That's brilliant indeed (no pun intended). BTW does the illustration at the end have something to do with U-V coordinate mapping of 3D objects in 3D modelling applications?

Looking forward to the livestream

eigenvectors and eigenvales - i just had a flashback to my linear algebra class!

Great subject !!!

so hyped about the lie theory livestream

This is so beautiful. I found the equation beautiful and fascinating in the beginning, and find it even more fascinating towards the end of the video. In the beginning I thought, although the symbols we use are so suggestive that this must work, it is actually a miracle how the process of exponentiation retains its structure between the realm of matrices and of scalars. Like, when a caterpillar becomes a butterfly it undergoes a phase where its body (including its brain) is a completely liquified stuff of free floating cells, and still there are experiments showing that the butterfly can remember things it learned before. Towards the end of the video I understand better how the exp, taking on different shapes, keeps its role of translating between addition and multiplication, even if these operators belong to different mathematical structures. And I am no less fascinated how this all works out.

Dr. Penn low-key thrilled to have his 9 e^9's lined up after the 1/9!

Awesome... Awesome! Very beautiful!

This is surprisingly nice trick! Now I am wondering is there any application to this feature? Such as in the Machine Learning area?

Looking forward to the live stream

An observation, e^Tr(A) can never be zero or negative. So does that mean that e^A for any matrix A must always have a positive determinant > 0 , therefore e^A must always have an inverse ( must always be non-singular). Am I right?

I'm wondering if you could do a video on the connection between conserved quantities (Noether's theorem) and Lie algebras...

Make sure the Livestream stays uploaded to your channel in case I miss it. I really want to watch it!

Great video

Nice video! It reminded me of a trick that was used during a first ODE course: matrix exponentials could be rewritten as matrix polynomials because of Cayley-Hamilton. That e^A from the example (2x2) would have become a_1*A + a_0*I, where these coefficients could be determined using C-H. I’m not sure if it holds for other types of functions, but it probably should work for smooth, analytic functions or something. Maybe that’s an idea for a video! :)

I really want to see the livestream. I oftentimes don't see them until after they are done, however. Could you make a couple/few posts about it when it gets closer to time so there is a higher chance I will see it?

you should do more on lie theory

@25:28 This is phenomenal. Is the live stream posted anywhere?

ok, michael...you stating that lie algebras are some of your favorites is reason enough for me to stick through these...

Many thanks. Very interesting. Any suggestion wheer to find good books or courses on the subject with possible applications (my background is engineering)? Thanks again

Matrix exponentials? More like "A lot of potential for learning these videos hold!" Thanks again for recording all these lectures.

At 13:00, upper right is e^6-e^3. I suspect the second eigenvector has a sign error.

Using the Jordan Classification Theorem (in its complex form) it is fairly easy to prove this, since as any matrix is conjugate to its Jordan form, and the determinant of the exponential of both are therefore equal. Lastly, any triangular matrix the formula its obviously true, are therefore for the Jordan form too.

just also for the jordan normal form

1:00 ah, yes, this determinant/trace exponentiation relation makes some sense, considering that in a matrix's eigenbasis the determinant is the _product_ of the diagonal entries (which are its eigenvalues, and in that basis should be its _only_ entries)

Man, all I wanted was something more out of Lie algebra. I've used it to show equilibrium points of an ODE system... but I feel like matrix exponential could really provide me an upper bound for the trajectories of this ODE system as well.

The fundamental group of SL(n, C) is trivial, also it's not compact. Hence, drawing it as a genus 2 torus was kind of a lie, but I do like the intuition it sparks.

great video! a minor correction: as far as i remember, the trace of a product of matrices is only invariant under cyclical permutations of the matrix product inside. at about 19:15, you made a different permutation and said the trace would not change, which is true but only because your matrices were special.

Great presentation! I am not following how the SLn group is representing the double Torus (or the surface of some other "object"). Also, how does it relate to the homology groups (that count the holes in the shape)?

Seeing the calculation of the exponential of a matrix makes me wonder, have you ever heard about Functional Calculus? It's a quite interesting theory that allows you to calculate matrix functions much more easily without having to diagonalize the matrix, among several other functions, of course

Your pronunciation of "eigen-" is absolutely perfekt.

So could this be used to help find the natural log?

Matrix exponentials seem quite fun. I wonder if octonion exponentials have any interesting properties.

The same argumentation passes for Jordan normal form matrices instead of diagonal ones, and that would do the trick as every matrix has a Jnf.

Thanks for example of a Lie bracket

13:27 the right upper coefficient just changed sign when taking over from the previous board. A tacit correction of a previous error.

Is this also valid for complex matrices?

This is what Feynmann used in his Path Integral right???

Hi Michael; I really liked your video! Can you give me a feedback on what I will write below? This video was, what I would say, on standard math; As a young man/student I went to Mathematical High School Belgrade...one of few of that kind in Europe...However I did not like so much the time I spent there because sometimes we were given difficult problems with less theory than I expected. What's more I used to compete in math competitions which were even more nonstandard... Eventually I scored high on the admission test to the University which was quite standard... I also like from time to time non-standard problems but the emphasis should be on standard problems...(LIKE IN SCHAUM SERIES)... Can you produce some video on how to prove continuity or uniform continuity of some elementary functions or simple combinations of them... Regards, Pedja.

Does someone know where is it posible to find the stream about lie algebras?

the love for useless calculations obscures the beauty of the mathematical structure hinted at in this video

A question comes to my mind: How do we know that e^( a matrix ) can be expressed as the power series expansion of e^x for matrices . We know that the power series expansion is true on C but. However i don't know about the set of matrices. Did we define it like that or we have some proof?

We can use the same idea by triangulsable matrix And every n xn are coefficients in C Is triangulsable

In case matrix of A not diagonalizable how can we calculate exp(tra(A))

14:20 ... by messing with the minus sign are you not correcting an earlier mistake made at 12:53 in evaluating the top right entry?

This seems like a matrix form of the power laws of a^(x+y) = (a^x)(a^y), and when x = y, a^2x = (a^x)^2. As was mentioned, the trace is the sum of eigenvalues, and the determinant is the product of eigenvalues, so the trace living in the world of addition, and the determinant lives in the world of multiplication, and the exponential is the bridge between these two worlds.