How (and why) to raise e to the power of a matrix | DE6

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Cites Rounak Sarkar · 2021-04 · Mathematics Stack Exchange · on “What is $(\frac{d}{dx})^\frac{d}{dx}$?”

How I got interested:- Yesterday I watched this [this video] new video from 3blue1brown. And I realized that we can use the formula - $\textstyle\displaystyle{e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}}$ to almost anything. We can define matrix powers [this video] , matrix to the power of a matrix [LINK] , exponential derivative [LINK] etc. So I started wondering about what $(\frac{d}{dx})^\frac{d}{dx}$ would mean. My approach:- Obviously in order to use the previously mentioned formula, we have to write $(\frac{d}{dx})^\frac{d}{dx}$ as a power of $e$ . And we can do that very easily. $\textstyle\displaystyle{\left(\frac{d}{dx}\right)^\frac{d}{dx}=e^{\frac{d}{dx}\ln(\frac{d}{dx})}}$ Now if we use the formu

Cites Kyan Cheung · 2021-04 · Mathematics Stack Exchange · on “What is $e^{\frac d{dx}}$?”

At the end of this video [this video] , 3blue1brown suggests it is possible to take $e^{\frac d{dx}}$ . So, what does $e^{\frac d{dx}}$ equal?

Cites DeltaIV · 2021-04 · Mathematics Stack Exchange · on “How to prove that $\exp\left(\mathbf{A}\right)=-\mathbf{I}$ for given $\mathbf{A}$?”

I watched this nice video [this video] by 3Blue1Brown and I would like to prove that $\exp\left(\mathbf{A}\right)=-\mathbf{I}$ where $$\mathbf{A}=\begin{bmatrix}0 & -\pi \\ \pi &0\end{bmatrix}$$ The video actually proves this, but it does it in a very complicated way IMO. It first computes $$\exp\left(\begin{bmatrix}0 & -1 \\ 1 &0\end{bmatrix}t\right)\ \forall t \in \mathbb{R}$$ and then plugs $t = \pi$ in the resulting expression). Are there simpler ways? What is the simplest way to compute the eigenvectors of $\mathbf{A}$ ?

Cites John D · 2021-04 · Mathematics Stack Exchange · on “2 to the power of a matrix”

I was watching 3b1b [this video] and I was wondering does there exist a definition for matrix exponentiation? Or is this concept of "matrix exponentiation" not well defined since we do not have a Taylor Series $\forall n \in\mathbb N$ ? example: $2^\mathbf A$ or in general, $n^\mathbf A, \ n \in \mathbb N \land \mathbf A$ is a matrix. I know this is probably nonsensical, but even if it isn't, it sounds interesting for how it could be used.

Cites Rdch · 2021-04 · Mathematics Stack Exchange · on “About Editing Math Videos for Teaching”

Given the situation we are in (pandemic) I have looked for the best way to make math videos for teaching. I have tried with digital whiteboards, notes in LaTeX but they do not convince me at all. Browsing YouTube I found this channel that is dedicated to sharing math topics. His videos are lovely and I would like to do something similar with my students. Do you know what kind of programs this guy uses to make those kinds of animations? [this video]

Cites pdaranda661 · 2021-05 · Mathematics Stack Exchange · on “Definiton of powers and exponents”

I was looking at this video [this video] , and since I've been recently studying Complex Analysis, where the number to the power of an imaginary number tend to appear, there is the main doubt that I would like to ask: What is the most complete definition of what a number to the power of something (either number of matrix or whatever) is? It may sound trivial, but since when you study maths in high school you study that a number to the power of another is that number multiplying itself the number of times defined by the second number, and this changes completely when you study more linear algebra, complex analysis, ... I started to see the case of a real number to the power of an integer as a part

Cites Lucas Perasolo · 2021-09 · Mathematics Stack Exchange · on “Is the area of the parallelogram the solution set of this Differential Equation? [Illustration](https://youtu.be/O85OWBJ2ayo?t=1338)”

I know that the determinant of matrix describes a change of volume of objects in space. The exp of a Matrix yields another matrix, therefore the determinant of it should also describe some change in volume in space. In 2D, as seen in the link, is the blue area the set of all possible solutions of a given ordinary differential equation? [LINK] If so, is this the reason why det(exp(M)) = exp(tr(A))? Credit: Visuals by 3Blue1Brown Illustration [this video]

Cites user1286095 · 2024-02 · Mathematics Stack Exchange · on “Does the exponential of a function converge? What can we do with it?”

I’m in middle school (6th grade) and I have a question related to the exponential function (the teacher couldn’t help me): We define the exponential as follows: $$ \exp(t) = \sum_{n = 0}^{\infty} \frac{t^n}{n!}$$ For whole number inputs of $t$ , this corresponds to raising $e$ to that power. I think $\exp(t)$ converges for every possible input of $t$ , but I haven’t found a proof yet. If anyone can provide a proof or disproof then that would be helpful. I would also like to know if for some $\exp(t)$ where $t \in U$ , $\exp(t)$ will also be $\in U$ . However, these are secondary questions and you can ignore them (but do answer if you want). We also define $\exp(t)$ for complex numbers [LINK]

Cites user1286095 · 2024-02 · Mathematics Stack Exchange · on “How to visualize matrix functions?”

I’m a middle school (6th grade) student who is self-learning these topics [LINK] , including linear algebra, and this is another question I have (teacher didn’t know again): I recently came across exponentiating matrices here [this video] , which we basically define by plugging in a matrix $A$ into the series; $\exp(A) = e^A = \sum_{n = 0}^{\infty} \frac{1}{n!}A^n$ . This is when I learned about matrix functions . Matrix functions take in matrices as inputs and output matrices: $f: A_x \rightarrow A_y$ . The function $\exp(A)$ is probably the most noteworthy example of a matrix function (do correct me if I’m wrong). One of the beauties of functions is their visualizations; graphs, vector fields,

Cites Zpazod · 2025-01 · Mathematics Stack Exchange · on “How to give a matrix exponent to a real/imaginary number”

I watched this video [this video] of 3Blue1Brown (which I understood very little at the time of writing this post), but I really want to know, if that's possible, how can we put an exponent to a number $x\in \mathbb{C}$ such as $$x^{A}=y$$ where $y$ is any number, or matrix (I told you, don't know how that works), and $A$ a square matrix. And if that's possible, in which domains can we use them? Note : If anything I said isn't clear, consider asking me, I'm not very good at expressing what I'm looking for. (Asking as a senior year high school student)