e to a matrix

Calculating e^A for a matrix A, explaining what this has to do with diagonalization, and solving systems of differential equations

[gast]

I stumbled upon this 25 years ago in a dissertation and was puzzled by what it meant a matrix as exponent. Unfortunately there was no YouTube back then to give me the answer. Thanks for clearing another impediment before I can die.

[dougr.]

Exponential matrices are useful in solving relativistic wave equations. (Dirac equation for Hydrogen atom).

[sebastiancuentasjimenez]

I speak very little English, but I understood most of the explanation, seriously thank you very much from Bolivia

[randperson]

Love the linear algebra. Keep it up Dr. P.

[aaronsmith]

This is so cool, thanks! This is an idea I stumbled upon in college, and it's neat to see that it has real application in mathematics.

[TheKiltman]

In general, you can always find a polynomial cancelled out by your matrix A (the characteristic polynomial, or the minimal polynomial if you're lucky) and then you can do 2 things to make the computation of the matrix powers easier:
1) naive : use that to get a recursion relation between A^n and smaller powers of A, which you have already computed
2)smart: for all integer p, compute an Euclidean division of X^p by said polynomial: when remplacing X by A, the term which is a multiple of said polynomial just cancels out and you are left with a simple polynomial in A of degree < N if N is the size of your matrix; this works nicely if you have an explicit expression of your matrix A and the polynomial cancelled out by A

[benjaminbrady]

You don't need it to be diagonalizable though since you can always raise to a power by hand, it's just really hard to calculate without eigenvectors

[LeandroDelgadoVentrue_O_Lobo]

Hi Dr. Peyam! I started my Master Degree in the last tusday and because of it, I stayed with my eyes in papers and books in the last four days to undestand this kind of exercise and I found a way to solve it, but I did not completly sure about my solution until I find your video! Your video confirm all my conclusions about this topic, thank you very much!

[lukehatcher]

This is an absolute GOD TIER explanation. Thank you

[daddymuggle]

Very nice presentation. You might want to do something about a minor error at 8:07, where you said (correctly) "e^A = P e^D P^-1" but wrote "e^A = P e^D e^-1".

[antoniocampos]

Man, you answer all the questions that I've been looking for in books and, as always, I never find them .

[Eric-xhee]

I forgot about this then I found your video after confused about how to solve for the matrix power of an exponential in a quantum computing function. I love your channel keep up the great work!

[PuzzleQodec]

I remember that these matrix exponentials were very useful in computer science for modeling dynamic systems with one of more feedback loops. I wonder where I can find that mysterious Feigenbaum constant in here.

[siddharthjoshi]

Dr Peyam, we love your videos and they are quite informative and good refresher. I have one suggestion, when you refer to some previous video for some concept, you can add the link in info button or in the description.
Thank you for the lovely content.

[rishabhpandeyaroll.]

I love this enthusiasm for maths, you earned a sub sir. Keep up the good work.

[RaniLink]

Thank you so much! from a Tel-Aviv University mechanical engineering student! <3

[omegadelta]

this was a great video to help me in times of need, mainly right before the exam, thanks you for breaking down everything so adequately.

[kamilbizon]

Soon we will have matrix derivatives on this channel

[varunalur]

Thank you so much professor, this helped me a lot.

[Fightclub]

In general you can use the Jordan normal form.