What is the difference between these two operations?

[zheng89120]

In first year, I learned that PAP-1 = D.

Now, I am learning that S+AS = A(v).

(where the + sign is a dagger)

It seems like both of these operations diagonalizes the matrix A, but doesn't feel like they are the same thing, unless they are the same thing.

[micromass]

What is the context?? What is A in the first and the second??

I bet that A in the second case is only allowed to be hermitian, while the first case is more general.