What is the difference between these two operations?

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The discussion centers on the mathematical operations PAP-1 = D and S+AS = A(v), both of which involve diagonalizing a matrix A. The first operation is more general, while the second specifically requires A to be Hermitian. Participants express uncertainty about the equivalence of these operations and seek clarification on the contexts in which they apply.

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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.
 
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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.
 

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