A MATRIX INTEGRAL! 

My new channel for formal math courses: youtube.com/@TheHouseOfMathness505?si=ID1g_2aTc2JNAIZD If you like the videos and would like to support the channel: www.patreon.com/Maths505 You can follow me on Instagram for write ups that come in handy for my videos: instagram.com/maths.505?igshid=MzRlODBiNWFlZA== My LinkedIn: www.linkedin.com/in/kamaal-mirza-86b380252

This is insane! I've never seen an integral so beautiful!! 😍😍

This is useful for solving linear time varying systems! Very interesting stuff!

I can watch your videos all day and never get tired of them. I love how you power through these amazing integrals with such ease. Thank you!

I feel like that solution was screaming to be cast in terms of sinh and cosh...

It was well detailed and well explained. You really gave us the best procedure to exponentiate a matrix. I enjoyed the video from the beginning to the end. It was awesome❤

gawd dayum bro, never seen an integral and matrix together😂, u got some wild questions, keep up the good work❤

One of the most integral I have ever seen

i suppose you could get the same result doing the calculation of e^A more straightforward, the only thing to notice is that A² = I, and after that the series becomes the following: e^(bA) = I • (1 + b²/2! + b⁴/4! + ...) + A • (b + b³/3! + b⁵/5! + ...) = I cosh(b) + A sinh(b) = ([cosh(b), sinh(b)], [sinh(b), cosh(b)]). or even without hyperbolic functions, just having noticed that e^b + e^(-b) = 2(1 + b²/2! + b⁴/4! + ...) and e^b - e^(-b) = 2(b + b³/3! + b⁵/5! + ...) but I should admit that the approach of finding the diagonalization of a symmetric matrix is more universal

Thank you for your effort.

Alternatively we can calculate the powers of the matrix M directly. Letting z=x^2 and A = ((0,1),(1,0)) we have M = z A. Now A.A = 1, the unit matrix U. A^3 = A etc. Hence the sum defining e^M = sum_0^inf (M^k/k! ) can be written, spliting even and odd k, as e^M = sum(m=0^inf) M^(2m)/(2m)! + sum(m=0^inf) M^(2m+1)/(2m+1)! =U sum(m=0^inf) z^(2m)/(2m)! + A sum(m=0^inf) z^(2m+1)/(2m+1)! = U Cosh(x^2) + A Sinh(x^2). Hence the integral of e^M is the Matrix of U int Cos(x^2) = U sqrt(pi)/4 (erf(x)+erfi(x)) and A int Sinh[x^2] = A sqrt(pi)/4 (-erf(x)+erfi(x))

Pretty interesting

Beautiful integral

This was quite challenging question

typeo on bottom row on frame 9 minutes and 29 seconds M(x y)transpose would equal (y x)transpose ie the order swaps since M = the all_ones matrix minus the I matrix

Cool vid! that makes me wonder, is it possible then to define a sort of gamma matrix function?

That was nice

Of course, things get much spicier if the eigenvectors of the matrix also vary with x.

One neat, equivalent method would be to convert to dual numbers and integrate e^(-jx^2)

hey maths 505: solve the genralized integral: intgeral from 0 to infinity ( e^ (-a x^2) multiplied by ln(nx)) dx

Mathematicians be like "Let me abuse this notation".

hi maths 505 could you tell how to prove reiman zeta(4) = pi raised to the 4th power/90?

You have a serious error at 11:24. You are missing e^(1/2) and e^B and e^B^-1

Babies who haven't took linear algebra: "OhhHH tHiS iS So HaRdCoRe!!!" Giga Chad Zeta(3) males who took linear algebra at least 4 times: "hun, I never thought Math s 505 posts trivial questions..." Great Idea to mix the bunch! gunna go hunt me down the same concept but with my favorite integrals from the channel ;P see how "contour integration" expands to them bad boys! - a random thought: if it is a 2x2 matrix I think we can use isomorphisms to turn them into complex variable functions which we're quite familiar to integrate with (since Isomorphisms have that nice "linearity" ) 12:57: if we insert the half inside... don't we get [cosh(x^2), -sinh(x^2), -sinh(x^2), erfi(x)]?

Ok so my question is can we do e^Mx = root ln(e^ln(Mx)), so the base of ln is e so the whole e term will disappear or is that wrong :(

12:43 that was an erf(x) not erfi(x)

saw this in a QM course

Can i get the course for this i mean advanced calculus

Is these problems filtered in your country or is it general to everyone

This was wild 😢😢😢😢

I did it with a different approach whilst eating aloo sabji pluss chawal By manipulating the series of e^x I wish we didn't get e^x² and only get e^-x² and then integrate from 0 to infinity to get beautiful sqrt(π)s everywhere

I have an integral im stuck on, it is the integral from 0 to inf of (e^-x^2 - e^-x)/x

PLEASE READ Can you fix your playlists such that the oldest videos are first, and the newest videos add to the end?

exp(B*A*B^(-1))=B*exp(A)*B^(-1)

Erf!

No I am not.

I'm joking left