Me: "Is this even possible?" Dr. Peyam:"What am I, an amateur?"

One of calmest and coolest person I have never seen. Fantastic explanation. Thanks a lot Dr.

Inverse function of the factorial

@Tanvir_Ahmed_Earth

Жыл бұрын

Can @DrPeyam do this?pls do this

@scooberdoom3502

Жыл бұрын

@@Tanvir_Ahmed_Earth it's such a cool question. Best answer I could find involving sterlings formula with a lambert W popping up inside. No idea how to approach it for a matrix equation but I'm quite sure, if anyone, our man is capable

LOVE YOU DR. PEYAM!! I am studying Mathematics at Imperial College London and your videos really help me get through the difficulty of the course. Thank you so much! Keep up the great content

wow, cool edits!

This just looks like pure wizardry to me as I don't know linear algebra 😂

Editing cool and smooth, professor!

@drpeyam

Жыл бұрын

Thank you!!!!

@theproofessayist8441

Жыл бұрын

Yeah those transitions were really smooth - finding eigenvectors is just tedious row reduction so we can skip that!

This is like every "what are eigenvalues and eigenvectors?" video but in reverse, and it's pretty nifty that you made it work. Like, I remembered that you've gotta use them to calculate matrix exponentiation, but I didn't remember how or why. And now I do. In about 5 minutes. Pretty sure that was about 2 full weeks of Linear Algebra you just refreshed for me.

Good. The exponential of matrix is applied to ODE. u'( x ) = - ( log 2 )u( x ) + 2( log 2 )v( x ) v'( x ) = - 6( log 2 )u( x ) + 6( log 2 )v( x )

@drpeyam

Жыл бұрын

Oooh that’s a nice application with the log(2)

Binary matrices are commonly used for encryption.

Lovely as always! Again for those rusty in linear algebra. Diagonal matrix form A=PDP^-1 where P is orthogonal matrix is made up of columns of your eigenvectors and D is diagonal matrix whose diagonal entries are the eigenvalues and everything else is zero. Stuff prior that is solving for characteristic polynomial to find eigenvalues and then plug them in and solve for when the matrix A-\lambda*(Identity Matrix) maps to the zero vector - then just lots of row reduction. *the yucky part for me - just my opinion lol*.

matrix exponentials are also used to test the stability of LTI control systems

@oximas-oe9vf

Жыл бұрын

What's an LTI control system ? If you may explain please

@vidhanp482

Жыл бұрын

@@oximas-oe9vf a physical system whose change in state i.e like change in position depends only on the previous state. If these quantities vary linearly and don't vary with time( only due to the previous state) they are linear time invariant, more formally any physical system whose laplace transfer function can be written as a proper fraction of two polynomial transfer functions.

Lovely video as usual Professor. Matrices are my happy place 😊🤍💭🔡

Should show that with Jordan form (and with any analytic function for that matter) to make it more general + work with any matrices - including non diagonalizable ones.

Nice video, also a nice exercise to work out from the more general context of functional calculus/spectral theory on operator algebras

Matrices in general behave like a scalar value within a special context; that is, when the matrix is operating on some vectors called the Eigenvectors of that matrix assuming that such exist for that particular matrix. So the idea of raising a scalar to another scalar is well defined in algebra. And the idea is even less strange if the initial problem is converted in terms of a logarithmic equation.

e^x can be defined whenever x is an element in a Banach algebra

Sir, you are inspirational. Everyone who have some proximity to mathematics will benefit a lot from you. Thank you.

@drpeyam

Жыл бұрын

Thanks so much!!

I like the way you present yourself

Really nice, we feel your passion

Loved it! 😇💐

Wow that's interesting😮.... You always come up with great content💖

Thanks Dr Peyam.

Suprised to see such a freindly matrix.

this is really cool, i like it!

me encantan tus videos , te felicito

Matrix exponentials are extremely useful in the field of statistics statistics. The density function of the multivariate Gaussian distribution has one of these babies in it.

@locallyringedspace3190

Жыл бұрын

In that case, the positive definite matrix generates an inner product (mapping vectors from the sample space to R) - so the exponential is scalar-valued. This is important in a lot of proofs since the scalar exponential is monotonic while the matrix exponential is not.

Always excited to watch your videos 😍😍 Love from India

SIR, YOU ARE THE BEST IN THE WORD!

That reminds me, in geometric algebra a multivector tranform of form uMu⁻¹ is link to the notion of rotor spinor algebra and lie algebra, that explain why the exponential function is the heart of algebra .

This seems to be the general process of performing operations with matrices. Could this be generalised, for a diagonalisable matrix A=PDP^-1, f(A) = P f(D) P^-1 If so, What restrictions do we have for f?

@ammaral-hakim3313

Жыл бұрын

f should be analytic

@unknownhero6187

Жыл бұрын

I didn’t get why it is allowed to do transition for applying the operation to the elements of the diagonal matrix? What is the idea behind this?

@alexandruandercou9851

Жыл бұрын

@@unknownhero6187 Taylor or MacLaurin series. Acording to these an analytic function can be writen as f(x)= sum i->infinity fdi(0)*x^i/i!. If you put a matrix inside f ,f(A) =sum fdi* A^i/I!. A^i=(P*D*P^-1)^i =P*D^i*P^-1 eg. A^2=P * D* P^-1 *P *D * P= P*D^2*P^-1 f(A) =sum fdi(0)* P * D^i/ i!* P ^-1 = sum P* ( fdi(0) * D^i / i !) *P^-1 = P* (sum ( fdi(0)*D^i / i !) ) * P^-1 = P * f(D)* P^-1 D^I if D is diagonal has on the diagonal D(a,a) ^ i and 0 everywhere else , if you multiply each element of D^i with fdi(0) and 1/ I! You get on diagonals D(a,a) ^i *fd(0)/ i ! . Call this matrix D(i) f(A) = P * (sum over i of D(i) ) * P ^-1 Last step is to perform the sum, the resulted value sumDi has sum fdi(0)*D(a,a) ^ i/ i ! on the main diagonal which is just f(D(a,a)) . f(A) = P* sumDi * P^ -1. For a 2×2 matrix sumDi =[ f(D(a,a) 0 0 f(D(a,a) ] note: fdi means the i th derivative of f.

@severoon

Жыл бұрын

Hmm KZread needs to support LaTeX. =D

@theproofessayist8441

Жыл бұрын

@@severoon Hell yes

Eigenvalue decomposition of channel matrix is needed all the time for Wireless channel..

i love Dr Peyam

I am once again asking for your video on the Matrix-th derivative of a matrix. I think it is possible… . . . Applying the Gamma function to diagonalizable matrices and using a rotation matrix as f (for example).

You are so happy doing math! That's how I was when I finally "got it".

very satisfying

This was a treat, I had seen the matrix exponential for e but hadn't though about other bases. So there's the matrix representation of complex numbers, just a first thought but could we diagonalize the matrix for i and then give a similar argument for i^i using matrices? thank you Dr. Peyam!

@drpeyam

Жыл бұрын

I think I did something similar in my playlist! Indeed you get a real matrix in that case

notice u can chech a^B=C by doing a^(TrB)=Det(C) in this case 32

Curious 👏♥️

Luv these powers to a matrix great use of linear algebra . any applications to random matrices or quantum mechanics ? eres el padrino!!👍

@KabeloMoiloa

Жыл бұрын

QM: well... solving the time-independent schrodinger equation is determining the eigenvalues of a matrix in the finite dimensional case or the spectrum of an infinite dimensional operator. The time dependent equation is basically saying that if you can diagonalise H then you can diagonalise exp(-it H) and in the eigenbasis you a diagonal operator that acts as |E_n> →exp(-itE_n)|E_n> where E_n are the eigenvalues/energies.

Dr payem!!!! my lovely crazy mathman😆😆😆😆

jamás me hubiera preguntado algo asi como un numero elevado a una matriz..... gracias por ese video

Great !!!!!!

This is so interesting! I recently watched 3blue1brown’s video on raising e to the power of a matrix, and in that video he said that e^A was basically shorthand for plugging A into the Taylor series for e. Would this method of computation work for computing e^A as well? This method seems a lot less cumbersome then figuring out what an infinite series of matrices converges to haha

@drpeyam

Жыл бұрын

Of course!

1:18 nice editing

As someone who hasn't taken linear algebra What is this wizardry

Man, math is so great

I'm trying to find more articles about the power of a Matrix, but I can't find a detailed article about this with more examples. Can you recommend articles and e-books with more examples about this?

I learned that method in a transmission technology lecture but don’t remember the application.

Can you upload a new video like this every day? Thanks a wounderful channel)

Can a function be taken to a matrix derivative?

Very easy to solve : 1. First find out the determinant value of the matrix. 2. Now raise 2 to the power of that determinant value

@drpeyam

Жыл бұрын

No?

Hi Dr. Peyam!

Exponential of a matrix is well defined because exponential of a linear operator has a convergent series due to operator norm. Then how do you define this 2 raised to power of a linear operator?

@Learnerz_isle

Жыл бұрын

The exponential of a matrix comes from its series expansion. So if u take 2^x, its maclaurin series would contain ln2 terms. The convergent may not be a issue as per I think, but i guess the series won't be the same.

At 1:25 there is a small error in the 2nd matrix, 6 - 3 = 3 not 4.

I checked using Julia language and it worked fine.

Next Time can you try matrix Power matrix?

@drpeyam

Жыл бұрын

Already done :)

Dr i I'm having a question regarding non homogeneous second order differential equations What will be the particular integral if second order differential equation get equal function with natural logarithms

@drpeyam

Жыл бұрын

?

thanks. video mentioned in 3:05?

@drpeyam

Жыл бұрын

It’s in the playlist :)

I didn’t get why it is allowed to do transition for applying the operation to the elements of the diagonal matrix? What is the idea behind this?

@drpeyam

Жыл бұрын

Check out my eigenvalues playlist :)

@joaobaptista4610

Жыл бұрын

The Caley-Hamilton theorem. Check for any linear algebra textbook that covers functions of square matrices.

2:32 why thats true ????

3Blue1Brown did 'e' to the power of a matrix!

@drpeyam

Жыл бұрын

I did that too :)

that is so cool ! i love this video! but i don't reallly know how to diagonalize a matrix :/

@buttercantfly2181

Жыл бұрын

and i know how eigenvalues and vectors work but i don't know why

@drpeyam

Жыл бұрын

Check out my playlists

Un vrai génie malheureusement on l’utilise pas en mécanique quantique :/

Why do you define the Characteristic polynomial of a matrix by det (A- λI) rather than φ(λ)= det (λI - A) ? I know the difference is just a signal (-1)^n and it is meaning less . But the definition is standard and the characteristic polynomial is Always a unitary polynomial (i.e. the coefficient of t^n is always 1) .

@drpeyam

Жыл бұрын

It depends on the textbook, some define it one way others define it the other way

exp{A} = / Jordan matrix / = ...

What happens if the matrix is not diagonalizable ?

@drpeyam

Жыл бұрын

Jordan form

Man i feel so behind. Used to talk with these singaporean dudes and they were saying they were doing this like freshman or sophomore year lol

why when i write this into calculator I get [0.5 4 1/64 64]?

@drpeyam

Жыл бұрын

The calculator calculates it incorrectly

@eckhardtdom

Жыл бұрын

@@drpeyam Well for a moment I thought calculating with matrices is easy :) It looks like it takes a little more progress to calculate it :)

Is there a use for this? Applied math? We don't need applications where we're going!

If y = m x + c Then y cos(θ) = x sin(θ) + c cos(θ) Where θ be any angle and c = intercept

@drpeyam

Жыл бұрын

?

@adrienanderson7439

Жыл бұрын

Do you write this because you could write the slope, m, as sin(θ)/cos(θ), where θ is the angle that the line would make with the positive x axis? So y= x (sin(θ)/cos(θ)) + c => y cos(θ) = x sin (θ) + c cos(θ). Like you could define the slope of lines using an angle and these trig functions rather then a number for rise over run.

electroboom?

Clp

Girls when they a Bad B*tch: 💄💇‍♀️💃💅 Boys when they a Bad B*tch:

🙃

my brain hurts

So this is how they got Tate.

Dude why HDR in a math video lol it hurts my eyes

This is impossible... my calculator says error: bad argument types

@kabivose

Жыл бұрын

Don't believe your calculator - mine can't factorise (10^100) +1

행렬은 문제로 내면 쉽지? 엑셀화해봐라

What the heck is a Matrix

Hello, every beoble:)

Did you get the jab?