Question: eigenvector equation discrepancies

Below is an eigenvector equation for  for the eigenvalue -3.4376...

When I solve the system below, I expect to get the vector

, just like the result of the Eigenvectors command used above.

Instead I get the zero vector.

In addition, when I compute the reduced row echelon form, I expect to have one of the rows be zero since matrix S has rank 1.

Therefore, I guess my question is maybe about calculations and not about Maple. That being said, I did the calculations manually, then I asked chatgpt, which agreed with me.

Why are the calculations not coinciding with Maple?

I can do the calculations "manually" with Maple:

Now, if I try to solve them simultaneously, it seems I have the same issue as with LinearSolve

I tried something else that worked, but the questions above still remain.

We start with  again

 =

In my problem I have the variable

 =

and it turns out that  is an eigenvalue of .

In particular, for  we get the values I showed previously.

For example, here is the eigenvalue we were looking at before

Okay so the eigenvector equation without subbing in a value for  is .

Here is what the matrix on the left looks like

And here we solve for the eigenvectors

Which is the expected result.

So, Maple can solve the eigenvector equation in this more general case but not if I specify  before asking it to solve.