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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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*} …
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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 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…
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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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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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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?
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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…
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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…
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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…
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Prove that $\displaystyle\lim_{k \to \infty} \left( I + \frac{1}{k}A \right)^{k} = e^A$

I'm having a little trouble here to prove the following statement: Let $A$ be an $n \times n$ matrix (real or complex). Prove that $$\lim_{k \to \infty} \left( I + \frac{1}{k} A \right)^{k} = e^{A}$$ Now I'm using matrix and possible…
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Prove that $e^{t(X+Y)}=e^{tX} e^{tY}$ implies $[X,Y]=0$

I am currently reading about the Baker-Campbell-Hausdorff formula and in a textbook on Lie Algebras, he shows that if $$[X,[X,Y]] = 0 \quad \text{ and } [Y,[X,Y]] = 0$$ then $$e^{Xt}e^{Yt} = e^{Xt + Yt + \frac{t^{2}}{2}[X,Y]}.$$ where $[X,Y] =…
JessicaK
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Does $e^{AB}=e^{BA}$ imply $AB=BA$?

Let $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times n}$. Is it true that $e^{AB}=e^{BA}$ implies $AB=BA$? If not, can you provide a counter example?
pulosky
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How do you find (continuous) bounds on the matrix exponential

Let $A$ be a $n \times n$ real or complex matrix. I am interested in bounds on the matrix exponential $e^{A t}$, for $t \geq 0$. In particular: is there a continuous function $C: M_{n\times n} \rightarrow \mathbb{R}_+$ such that $$\|e^{At}\| \leq…
Theone
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Calculating matrix exponential

Given matrix $$M = \begin{pmatrix} 7i& -6-2i\\6-2i&-7i\end{pmatrix}$$ how do I calculate matrix exponential $e^M$? I know I can use that $e^A=Pe^DP^{-1}$ where $D=P^{-1}AP$. I computed the characteristic polynomial of the above matrix…
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Finding the solution to a non-homogeneous matrix exponential.

Consider $$\frac{d}{du} \begin{bmatrix}a\\b \end{bmatrix} = \begin{bmatrix}-x& y\\ -y&-x\end{bmatrix} \begin{bmatrix}a\\ b\end{bmatrix} + \begin{bmatrix}\cos(zu)\\ -\sin(zu)\end{bmatrix}$$ where $P = \begin{bmatrix}-x& y\\ -y&-x\end{bmatrix}$.…
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