# Eigen Decomposition A real symmetric matrix is said to be orthogonally diagonalizable. A matrix is called symmetric if $A = A^T$. An orthogonal matrix is a square matrix for which $A^{-1} = A^T$. An orthogonally diagonalizable matrix $A$ can be decomposed as $ \boldsymbol{A} = \boldsymbol{U} \mathbf{\Lambda} \boldsymbol{U}^{T} $ where $\mathbf{\Lambda}$ is a diagonal matrix of whose diagonal elements are eigenvalues of $\mathbf{A}$ and $\mathbf{U}$ is a change basis matrix, whose columns as eigenvectors of $\mathbf{A}$. This equation is called Eigen decomposition of $\mathbf{A}$. --- ## References