Spectral Theorem

Description:

• Any symmetric matrix is orthogonally similar to a diagonal matrix. This is stated in the following so-called spectral theorem for symmetric matrices
• theorem 1:
  • Let $A\in {\mathbb{R}}^{n,n}$ be Symmetric Matrix. Then there exist real numbers ${\lambda }_i\in \mathbb{R},i=1,\dots ,n$ and a set of orthonormal vectors $u_i\in {\mathbb{R}}^n,i=1,\dots ,n$, such that $A{u}_i={\lambda }_i{u}_i,i=1,\dots ,n$
    • The numbers ${\lambda }_i,i=1,\dots ,n$ are the eigenvalues of $A$ and vectors $u_i,i=1,\dots ,n$ are associated eigenvectors.
  • Equivalently, there exist an orthogonal matrix $U=\left[u_1\cdots u_n\right]$ (i.e., $U{U}^T=U^{T}U=I_n)$ and a diagonal matrix $\Lambda =\mathrm{diag}\left({\lambda }_1,\dots ,{\lambda }_n\right)$, such that $A=U\wedge U^T={\sum }_{i=1}^n{\lambda }_i{u}_i{u}_i^T,\Lambda =\mathrm{diag}\left({\lambda }_1,\dots ,{\lambda }_n\right)$
  • The theorem says: for a symmetric matrix, all eigenvalues are real, and eigenvectors can be chosen to be mutually orthogonal