How does the function MatrixExp[A] in Mathmatica works?

Questions
1

Dear Sir or Madam:
I wonder how the function MatrixExp works.
Does MatrixExp(A) refer to calculate
1.png
?

For example, A = {{1.2, 3.7, 5.6}, {0, -2.2, 4}, {-5.3, 0, 1.4}}.
Firstly, I used the textbook method to compute
1.png
. The result was shown as below.

Open in Wolfram Cloud ↗

Then I computed MatrixExp[A]. The result was shown as below, which is different from the first one.

微信截图_20230215214235.png
微信截图_20230215214235.png340×206 16.5 KB

So what’s the problem?
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=1.png&userId=2729333
[2]: https://community.wolfram.com//c/portal/getImageAttachment?filename=1.png&userId=2729333
[3]: Wolfram Cloud Document
[4]: https://community.wolfram.com//c/portal/getImageAttachment?filename=微信截图_20230215214235.png&userId=2729333

2

That textbook method looks like it might be computing the exponential of A*t. There is no possibility that MatrixExp[A] will have some new variable in it.

Also it’s not a good idea to compute Eigenvalues and Eigenvectors separately since they might not correspond.

3

I got it. Thank you for your reply.