Orthogonal Eigen-Decomposition/Diagonalization is the diagonalization of a symmetric matrix 𝐴 which is defined as:

• 𝐴 = 𝑃𝐷𝑃 ^-1= 𝑃𝐷𝑃 ^𝑇

where:

• 𝑃 is a square orthonormal matrix; columns and rows of 𝑃 are the orthogonal eigenvectors of 𝐴 (thus: 𝑃 ^𝑇𝑃 = 𝑃𝑃 ^𝑇= 𝐼)
• 𝐷 is a diagonal matrix

Proof

• Spectral Theorem - 𝐴 is Orthogonally Diagonalizable ⟺ 𝐴 is Symmetric