Orthogonally diagonalizable matrix and orthonormal basis

Question

How is an orthogonally diagonalizable matrix related to eigenvectors and an orthonormal basis?

Answer 1

"Orthogonally diagonalizable" means precisely "has an orthonormal basis of eigenvectors".

Answer 2

We say that a matrix $A$ is orthogonally diagonalizable if we can write $A=D^{-1}SD$ such that $S$ is a diagonal matrix and its diagonal elements are eigenvalues of $A$ and rows of $D$ are eigenvectors of $A$. Rows of $D$ are a orthonormal basis, it means they or mutually perpendicular and because of this fact $D^T=D^{-1}$.

As an example you can find all symmetric matrices orthogonally diagonalizable.