# Questions tagged [matrix-exponential]

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

608 questions
Filter by
Sorted by
Tagged with
84 views

### Why does the matrix exponential $e^A$ always exist?

Why does $e^A$ always exist for any given $n \times n$ matrix $A$? I can't find anything discussing this question, which is quite suprising, since it is such a general question.
22 views

### Higher order derivative of exponential map

The derivative of the exponential map is given by (wiki): $$\frac{d}{dt} e^{X(t)} = e^{X(t)} \frac{1 - e^{-ad_{X(t)}}}{ad_{X(t)}} \frac{d}{dt}X(t)$$ Is there a reasonable formula for higher order ...
• 4,478
28 views

1 vote
66 views

### What is the correct Taylor expansion of exponentiated operators?

I am trying to derive the following result from this paper (Equation S3):$$e^{-ik_z\hat{z}}\hat{\rho}e^{+ik_z\hat{z}} - \hat{\rho} \approx k_z^2[\hat{z},[\hat{z}, \hat{\rho}]]$$with operators $\rho, z$...
79 views

### Dealing with exponential operators

I have some troubles with the following equations. First, we define a function $g\left(\hat{\vec{x}}\right) = \exp[-(\vec{x}-\hat{\vec{x}})^2]$. We want to expand $g$ around $\hat{\vec{x}}\approx0$. ...
67 views

### Prove that the exponential of matrix is well defined

We just saw today the concept of the exponential of a matrix. Let $A\in \mathbb R^{n\times n}$ a matrix. How can I prove that $$e^A:=\sum_{k=0}^\infty \frac{A^k}{k!}$$ is well defined ? What I've done ...
• 672
1 vote
42 views

• 1,551
1 vote
25 views

### Calculating an element $b_{ij}$ of matrix exponential $(b_{ij})=B=\exp(A)$

Is there some easy way to calculate an element $b_{ij}$ for fixed $i$ and $j$ of the matrix exponential $(b_{ij})=B=\exp(A)$? ($A$ is symmetric real matrix) I know that I could diagonalize $A$ and ...
• 359
75 views

### Matrix exponential via Cayley-Hamilton

Problem For any $t\in\mathbb{R}$ compute $\exp(A_\omega t)$, where \begin{equation*}A_\omega\triangleq\left[\begin{array}{c|c} 0_2 & I_2 \\ \hline 0_2 & \Omega \end{array}\right]\end{equation*}...
• 311
1 vote