Definition of orthogonally diagonalizable matrix

Definition. An $n \times n$ matrix $A$ is orthogonally diagonalizable if there exists an orthogonal matrix $P$ (i.e., $P^T = P^{-1}$) and a diagonal matrix $D$ such that:
$$A = P D P^T \quad \text{or equivalently} \quad P^T A P = D$$

This means $A$ has an orthonormal basis of eigenvectors. The columns of $P$ are the orthonormal eigenvectors of $A$, and the diagonal entries of $D$ are the corresponding eigenvalues.

Example. The symmetric matrix $A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$ is orthogonally diagonalizable with $P = I$ and $D = A$.

Note: Orthogonal diagonalizability is stronger than ordinary diagonalizability -- it requires not just a basis of eigenvectors, but an orthonormal basis.