All Questions
4
questions
1
vote
1
answer
87
views
Evaluate $\lim_{j \rightarrow +\infty} (I + A/j)^j$
Let $A$ be a $n\times n$ matrix. Evaluate
$$
\lim_{j \rightarrow +\infty} \left(I + \frac{A}{j}\right)^j.
$$
My guess is $e^A$.
My attept:
\begin{align*}
\lim_{j \rightarrow +\infty} (I + \frac{...
- 2,316
0
votes
3
answers
121
views
Exponentiation of a $2\times 2$ matrix
We know:
$$\exp(At)=I+ \sum^{\infty}_{n=1}\frac{A^nt^n}{n!}$$
Here $$A= \begin{pmatrix} 0 & 1 \\ -w^2 & 0\end{pmatrix}$$ is a $2\times 2$ matrix,
$I$ is identity matrix.
How to show:
$$\...
- 1,919
0
votes
2
answers
78
views
Let $A$ be a single $p\times p$ Jordan block. Find general solution to $\dfrac{dx}{dt} = Ax$
Let $A$ be a single $p\times p$ Jordan block. Find the general solution to $\,\dfrac{dx}{dt} = Ax$.
What should I approach first? Please help!
- 117
8
votes
4
answers
7k
views
How to compute time ordered Exponential?
So say you have a matrix dependent on a variable t:
$A(t)$
How do you compute $e^{A(t)}$ ?
It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied here ...
- 17.5k