Let $A \in \mathbb{R}^{n\times n}$ be symmetric with diagonalisation $A = P\Lambda P^{T}$, where $P\in \mathbb{R}^{n\times n}$ is unitary and $\Lambda = \mathrm{diag}(\lambda_1, \lambda_2, \dots, \lambda_n) $, where $\lambda_1, \dots, \lambda_n$ are the eigenvalues of A.
My question is, is such a diagonalisation unique? In particular, are diagonal entries of $\Lambda$ necessarily the eigenvalues of $A$?