Assume we have a matrix $H$ that is diagonizable. If we perform spectral decomposition on $H$ forming a new matrix $H'$ whose columns are the eigenbases of $H$, can we say that $H' =H$?
In other words, if two matrices share the same basis, eigenvalues, eigenvectors, and transformation operations, can we conclude they are the same matrix?
$$H=PDP^{-1}$$ where $D$ is formed from the Eigenvalues and $P$ from the Eigenvectors. Unless you swap these elements, the matrix is uniquely defined.