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Questions tagged [matrix-exponential]

"The matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function."

Where $X$ is a real or complex square matrix, $e^X \equiv \sum\limits_{k=0}^\infty \cfrac{X^k}{k!}$. $X^0$ is defined to be the identity matrix with the same dimensions as $X$. This is analogous to $e^x = \sum\limits_{k=0}^\infty \cfrac{x^k}{k!}$, where $x$ is a real or complex number.

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If $e^A$ and $e^B$ commute, do $A$ and $B$ commute for finite dimensional matrices?

It is known that if two matrices $A,B \in M_n(\mathbb{C})$ commute, then $e^A$ and $e^B$ commute. Is the converse true? If $e^A$ and $e^B$ commute, do $A$ and $B$ commute? Edit: Additionally, what happens in $M_n(\mathbb{R})$? Nota Bene: As a…
Seirios
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Integral of matrix exponential

Let $A$ be an $n \times n$ matrix. Then the solution of the initial value problem \begin{align*} \dot{x}(t) = A x(t), \quad x(0) = x_0 \end{align*} is given by $x(t) = \mathrm{e}^{At} x_0$. I am interested in the following matrix \begin{align*} …
samsa44
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How to prove $\det \left(e^A\right) = e^{\operatorname{tr}(A)}$?

Prove $$\det \left( e^A \right) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}^{n \times n}$.
John
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Derivative of matrix exponential w.r.t. to each element of the matrix

I have $x= \exp(At)$ where $A$ is a matrix. I would like to find derivative of $x$ with respect to each element of $A$. Could anyone help with this problem?
darwin rajpal
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Is there a general formula for the derivative of $\exp(A(x))$ when $A(x)$ is a matrix?

It's easy for scalars, $(\exp(a(x)))' = a' e^a$. But can anything be said about matrices? Do $A(x)$ and $A'(x)$ commute such that $(\exp(A(x)))' = A' e^A = e^A A'$ or is this only a special case?
Tobias Kienzler
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Developing intuition for Lie groups and Lie algebras

Background: Until now, I've been able to motivate everything I've learned in mathematics, and develop some solid insights for everything. But I learned some Lie theory this summer, and while I have a good grasp of the elementary aspects and strong…
Abhimanyu Pallavi Sudhir
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What is an interpretation of the matrix exponential?

I just read about the existence of the "matrix exponential" $$e^X := \sum_{k = 0}^\infty\frac1{k!}X^k$$ Is there a simple way to interpret this? I understand the analog between real number exponentials as infinite Taylor expansions. However, I have…
user56834
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Matrix exponential of a $3 \times 3$ skew-symmetric matrix without series expansion

I have the following skew-symmetric matrix $$C = \begin{bmatrix} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \end{bmatrix}$$ How do I compute $e^{C}$ without resorting to the series expansion…
Nirvana
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The matrix logarithm is well-defined - but how can we *algebraically* see that it is inverse to the exponential, as a finite polynomial?

This question is inspired by this which I saw earlier today. I started writing my answer, to share the insight that the matrix logarithm can be defined on matrices that do not have unit norm using an alternative technique. Now, Sangchul has posted a…
FShrike
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Given a unitary matrix $U$, how do I find $A$ such that $U=e^{iA}$?

A unitary matrix $U \in \mathbb C^{n \times n}$ can always be written in exponential form $$U = e^{iA} \tag{1}$$ where $A$ is Hermitian. My goal is to find the Hermitian matrix $A$, given the unitary matrix $U$. I figured out a way by diagonalizing…
WishBeLeibniz
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How to calculate the matrix exponential explicitly for a matrix which isn't diagonalizable?

How can I compute an expression for $(\exp(Qt))_{i,j}$ for some fixed $i, j$ and matrix $Q$? When $Q$ is diagonalizable, we can diagonalize, but what can be done otherwise? Thanks.
Ben Derrett
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How to understand the exponential operator geometrically?

Consider the geometric interpretation of an orthogonal matrix, a projection matrix, a (Householder) reflector, or even just matrix-vector multiply in general. A matrix takes a vector from a vector space (after a basis has been fixed) and performs a…
Fixed Point
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How to find the matrix exponential of non-diagonalizable matrix?

I am currently self-learning about the matrix exponential and found that determining the matrix of a diagonalizable matrix is pretty straight forward :). I do not, however, know how to find the exponential matrix of a non-diagonalizable matrix. For…
user860374
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Commuting in matrix exponential

Let $A$, and $B$ be commuting $n\times n$ matrices, i.e., $A B = B A$. Let $$ \exp(A) := \sum_{i=0}^\infty\frac{1}{i!} A^i $$ Show that $\exp(A+B) = \exp(A) \exp(B)$.
user25004
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Proving the Lie-Product formula

Let $A,B\in gl(n)$. Then prove $$e^{A+B} = \lim_{k\to\infty} \left (e^{\frac{A}{k}}e^{\frac{B}{k}}\right )^k$$ I found this theorem in this notes http://www4.ncsu.edu/~aalexan3/articles/liegroups.pdf but there is no proof. I am new to exponential of…
Kenneth.K
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