Eigenvalues and eigenvectors - uniqueness

Suppose I have a square $n\times n$ matrix A with $n$ linearly independent eigenvectors.

Clearly more than one matrix can share the same eigenvectors and eigenvalues.

However, I also know that I can write this matrix A in the form D = P$^{-1}$AP, where D is a diagonal matrix with diagonal entries equal to the eigenvalues and the columns of P are the eigenvectors of A.

However, in the other direction, if I know the eigenvalues and eigenvectors of a matrix A, then I can form the matrices P and D using the above. However, this seems to suggest that given eigenvalues and eigenvectors I can find a single matrix that corresponds to these values...

So when are eigenvalues and eigenvetors unique and not unique? What am I missing?

• You can have the same eigenvalues, or same eigen vectors, but each eigenvalue cannot be linked to the same eigen vector – Atmos Mar 9 '18 at 12:43
• Set the order in the set of eigenvectors, set the order in the set of eigenvalues, now you can generate the permutations in the selected set, the pairs of eigenvalue-eigenvector are important.. – Widawensen Mar 9 '18 at 12:53

If you know $n$ distinct eigenvalues and an eigenvector for each (not just the set of eigenvectors) for a linear transformation $T$ then those eigenvectors will be linearly independent and so form a basis. In that basis the matrix of the transformation will be diagonal.
If you change bases using $P$ as in your question then in the new coordinate system $T$ will have the transformed matrix. But it will still be the same transformation, with the same eigenvalues and eigenvectors.