symmetric matrix

Symmetric Matrix

A matrix $A$ is a symmetric if $A=A^T$, or equivalently $Q_{ij}=Q_{ji}\forall 1,\dots ,n$.
Symmetric matrices have real eigenvalues and orthogonal eigenvectors.
$⟹$ There exists a rotation matrix $U$ which rotates the basis to the eigenvectors of $A$.
$⟹$ This rotation is reversible by $U^{-1}=U^T$ $⟹$ $A=UD{U}^T$ where $D$ is a diagonal matrix with the eigenvalues of $A$.
$⟹$ Symmetric matrices are always diagonalizeable (spectral theorem).