orthogonal diagonalization

Diagonalizing a symmetric matrix using an orthonormal eigenbasis, so $P^{-1}=P^{\mathsf T}$ .

A real symmetric matrix ( $A=A^{\mathsf T}$ ) has only real eigenvalues, and eigenvectors from different eigenvalues are automatically orthogonal — no Gram-Schmidt needed between distinct eigenspaces. Assembling an orthonormal eigenbasis into $P$ gives $A=PDP^{\mathsf T}$ , since an orthogonal $P$ has $P^{-1}=P^{\mathsf T}$ by inspection, skipping the row reduction ordinary diagonalization needs.