A very interesting matrix integral
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- Опубликовано: 20 сен 2024
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1. It always fascinates me to define the transcendental functions on other rings, not only the ones we're used to. Functional analysis has to be in my top three of favorite fields in mathematics, the way you apply the logic of abstract and linear algebra to stuff from calculus opens the door to a buffet of amazing problems.
2. We're the ones who should be thanking you, this isn't just my favorite math channel, this is literally my favorite channel, it brings what I love the most and turns it into something we can call art. You provide something other channels, including math channels, don't. and it is this sense of amiability with the viewer. you aren't just uploading videos, you're making sure everyone that comes in here fully understands what you present.
moreover, It was amazing to watch myself grow as an aspiring mathematician along side this channel, I started from only dreaming about solving problems like these, to actually being able to solve them. I always feel like I have an extra edge from all the techniques you delivered on this channel.
so for that thank you, and here's to another year of growth and mathematics.
So may the lord bless you on this holy season, and hope you come back with more power!
Thank you for being a great friend ❤️ and it's always wonderful to read your comments and perspectives on solution developments. They keep getting cooler and cooler everytime.
I agree completely. One of the best math channels out there.
Nooooooooo I wanted the new year's upload. 🥺😭😭
I hoped you'd pull another crazy one off and introduce us to something new like you did last year with contour integration. But anyway.
Aw man, it has been a great journey so far. 2023 was absolutely crazy, and I loved watching every bit of this channel's growth, all the way from infancy to its current pubescent stage. Hopefully, I'll get to witness its grand leap into adulthood.
I have really appreciated you being my teacher this year, and I'm grateful for every bit of information you bestowed upon us. It never felt forced, and every bit of knowledge was masterfully presented.
I still remember it like it was yesterday when I started watching this channel back in December of 2022, and it just instantly clicked. It felt as if I was dying of thirst, but never realized it. The outflow of high-quality content was - and still is - absolutely impressive to say the least. Hell, I even made a quote back then stating that "It isn't a Maths 505 video if the solution development isn't elegant as hell".
This has been a fun ride so far, and I can speak for everyone here when I say that we're all looking forward to what's upcoming following the arrival of the new year.
2023 was stockpiled with great memories on this channel, and I hope that 2024 brings forth even more.
Here's to another year of prosperity, and to another year of Maths 505. 🌹♥️
Ciao brother, see you next year. ❤️
Thanks bro ❤️
I'm incredibly happy that I've made great friends like you because of the channel ❤️
The best math teacher on youtube
I think so😊
no
I discovered this channel a few days ago, it is already my favorite math channel - keep up the good work
its your fav 'math' channel but its my 'overall fav channel' in youtube
I really have the best subscribers ❤️
This channel is still solving a lot of challenging math problems!
Watching this channel can possibly prevent dementia!
I never even thought of integrating a matrix, let alone the cosine of one. This video blew my mind!
Thank you for everything! Every video is a gift.
This holiday special was insanely entertaining. Thank you so much for that.
Oh yeah. We did this in our functional ODEs course.
Merry Crimbo, thank you for all your hard work. Good luck in grad school!
I'm so happy that I've found your channel around a 9 months ago. Thank you for that incredible inttegtals and DE's, love it!
keep up the good work man, you really deserve how much your channel has grown and much more, love your vids
I can't tell how much of advance math stuff i have learnt from the channel by just watching your solutions, It was really a good year. Thank You!
Thank you for being here Kamaal. No one seems to cover math topics like you do, your videos always have a relaxed and curious vibe to them. It's always great to sit down after a day of studies and watch you get some beautiful results out of integrals while applying ingenious methods. You have a great way of transmitting information and I am really thankful for your content. After a day of university, your videos always help me regain my motivation to continue studying. I'd like to wish you a Merry Christmas, or whatever it is you may be celebrating. Hope to have another great year with you. All the best.
Thanks mate
This is such a valuable channel for studying integrals.
Thanks for publishing your videos, they are really helpful :D
I just recently discovered your channel. Your videos are so interesting and the problems you cover are always challenging. Without a doubt I know that your channel will grow even more next year. Happy holidays!
I love watching these wild integrals lmao
Bro should put the differential element inside the matrix one day 👀
Hi,
Thanks a lot for everything you have done, integrals and series fascinate me as much as you.
Happy New Year and see you soon.
Happy new year bro
So humble then ending was wholesome, Thank YOU for sharing
coooool! What a gem of an integral and what a gem of a youtube channel. Happy holidays to you!
Thank you too! I've learnt a lot of integration tricks from your videos, and watching you solving integrals is pleasurable
Thank you so much for the super fascinating content you put out!
I love the way how solve it keep posting such a thing, happy new year and thank you so much for the beautiful content
I can’t say it enough. I love your videos.
You are the reason why I like calculus and math in general. Thank you Math 505, you made me see math as something more fun
Congratulations on an awesome year of growth and thanks for keeping us entertained with your constant uploads. Happy to see you grow from success to success. Happy new year and see you in 2024!
Happy new year mate
Isn't the integral of 0, C? When you integrated the matrix, you made all of the zero entries 0 instead of some C.
the First Time i ever see a Matrix inside an integral sign, this was really fun
I would have diagonalized matrix A then distributed the scalar and the integral operator.
Are you allowed to put A^0 =I_3 given that A is not invertible (det A =0)?
Greetings sire! Your videos are highly entertaining and especially some integrals are extremely interesting especially when it comes to watching the long solutions end with a nice constant. I must ask though, what application do you use and is it free? Where can I download it from? It would be much superb if you could by any chance provide me such an information. Thank you in advance.
Sincerely,
loving the videos bro, happy new year!
Happy new year
this is my first video of yours, loved it
If x^2 = u, that actually implies x = +/- sqrt(u).
What’s the implication of not considering -sqrt(u)?? Seems incomplete when this is not considered
As always, fun, surprising and nicely done. Happy Holidays.
This is one of the main reasons I like math. You can do absolutely nonsensical things like putting a matrix into a trigonometric function into an integral and you somehow can apply perfectly reasonable logic to get a wierd but completely normal result
Do an explicit calculation with the Haar measure of the orthogonal group (integrating with respect to a matrix group)
cos and integral both are operator act on the matrix but for cos you do that trick but for integral you take it individually from each element. I don't know why this discrimination happened. Could you please make it clear?
Don’t know much about integrating matrices or matrices themselves. But if an integral of a matrix is the matrix with that integral of all of its components. Then in the case of this example, wouldn’t you integrate all the 0 components, giving you a different selectable constant for each of the zero components?
This is a definite integral.
@@maths_505 oh yeah. Duh
To say that I followed along would be a lie, but it's always good to find the limit of your understanding to guide you to further knowledge when you're ready to pursue it. So thank you! I'll go look up "what does Gamma mean?" tomorrow, lol
Thank you for your work sir!
A matrix is somewhat structural. And a wave impact of cosine.
Thank you for your videos!
Stupid question: I understand this is a definite integral and therefore the constant is not important because it cancels, however if you were to make an indefinite integration of a matrix, would that imply that the same constant goes into every element of the matrix? I would assume so, but this implies that although the indefinite integral of a matrix is indeed equal to the indefinite integral of each of its elements provided they all have the same constant, the reverse is not true if no information is given about the constants in the indefinite integrals inside a matrix?
A better way to look at this is the constant of integration is a constant matrix.
@@maths_505 Well obviously, d'oh. :D Great answer! And of course that matrix would have c11, c12, c13, c21, ..., c33 in the example of a 3-by-3. Got it.
@IshayuG and I didn't think the question was stupid. I'm actually surprised no one asked this yet.
I apologize if my English is not very clear, as I am Japanese.
This integral was very interesting, and I made some new discoveries.
By the way, I have a favor to ask.
I would appreciate it if you could discuss integrals related to hypergeometric series and elliptic integrals.
The integrals you're working on are of a very high difficulty, and I find them quite fascinating as well. (Of course, I'm subscribed to the channel.)
I would like to delve into highly advanced integrals related to hypergeometric series and elliptic integrals.
So, I'm hoping you could handle that for me.
Thank you in advance!!
Elliptic integrals are an area of interest of mine. I haven't gone too deep into hypergeometric series but I plan to soon.
And the English was perfectly clear don't worry.
I’m looking forward to it!
What’s the program you’re using? Looks crazy useful.
i wish you merry Christmas and happy new year
Love you buddy ❤ from Morocco
This is something straight out of a maths nightmare 💀💀
This actually is very cool
The good teacher needs good students
what's a matrix
4:20 dx
Thats a lot of problems with definition of integral of a matrix. Is it correct in this video? To check it, find the derivative, but using definition. Sometimes it gives a wrong answer. For example, in systems of ODE, general matrix Ф is a kind of integral of exponential matrix A, but integrated in special way, such that Ф’ = AФ. And it causes the result, that sometimes integral of a matrix is not a matrix of integrals. Keep it mind.
Talent Talks 😉.Thank you
How did you multiply the matrix? What steps did you do?
keep going pls i love your videos
Math 505 - Sir, from now on you’ll be Home Thompson from Terror Lake. Let’s just practice a bit. When I say Hello Mr Thompson, you’ll say Hi.
Math 505 - Hello Mr Thompson.
Me - …
great video, I have just a doubt: what allows us to take Taylor series of cosine near x=0 when the integration domain is till infinity?
Thanks, and keep going with those videos!
It only requires two things.
The function itself be analytic in sense of complex analysis, and the coefficients of the power series decay super-exponentially to zero.
The radius of convergence can be proved infinite by root test.
@@pnintetr so we could have taken the center of the series in another point (for instance x=pi/2) and get the same result? Or the point x=0 has a preferencial meaning? Thanks in advance
Cool vid.
Do you know any applications of matrix integral?
Applying functions to matrices and integrating matrix fields often occurs in quantum mechanics.
What app do you use to write your notes here?
Isn't the integral of 0 a constant? Didn't get this part
These are definite integrals.
@@maths_505 hmm gotta study more
You’re losing the plot with these crazy integrals 😭 super creative ideas
Why can we take the integral sign inside the Matrix? What is this called?
That's how integration of a matrix field is defined.
@@maths_505 ah ok gotta go through it then
Every physicist know how to integrate and differentiate operators...
For the x^2*exp(-x^2) integral, it's easier to use integration by part
What is the program?
Out of curiosity, what type of real-world scenario would a problem like this represent?
I haven't seen integrals like these in my physics studies so far but defining functions of matrices is a common occurrence in quantum mechanics and functional analysis.
@@maths_505 Thanks! Generative AI has me re-kindling my love for math and wanting to learn more linear algebra.
www.diva-portal.org/smash/get/diva2:731610/FULLTEXT01.pdf%22
Theyre used in theoretical physics.
Maybe in a linear system of Differential equations like xdot = iAx the real part of the solution is cos(At) * constant vector but idk where this is helpful. 🤔
Hello,
I've been watching some of your content and I wanted to ask what textbook do you recommend for starting in complex analysis?
Complex analysis by Gamelin
@@maths_505 Thank you
any utility for this?
Wolframalpha gives a different answer
Is it wrong or are you wrong?
I dont know, but one of y'all is not correct
very nice
amazin
♥️♥️♥️♥️
Surely You're Joking?!!
Your voice is so stressed, feels like you recorded this video at 3am.
Not stressed. Just tired....it was 3 am😂
@@maths_505 seriously! Finally I passed the probability exam. 🙂
♾👏🏻🎉
Isn't this a bit artificial?
❤😂🎉😢😮😊😅😮😢😢🎉😂
So elitist lol
Never been this early
kng
👍👍👍
I like matrices showing up in unusual places. Cheerful Calculations 🧮!