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# Definition of orthogonally diagonalizable matrix

Put content here**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:
$$A = P D P^T \quad \text{or equivalently} \quad P^T A P = D$$
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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.
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**Example.** The symmetric matrix $A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$ is orthogonally diagonalizable with $P = I$ and $D = A$.
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Note: Orthogonal diagonalizability is stronger than ordinary diagonalizability -- it requires not just a basis of eigenvectors, but an *orthonormal* basis.

# Parents

* Symmetric matrices