Spectral theorem

In linear algebra, the spectral theorem states that for symmetric matrices, i.e., $A=A^T$, then:

• Every symmetric matrix is orthogonally diagonalisable, i.e., there exists an orthogonal $S$ such that $S^{-1}AS=S^{T}AS$ is diagonal.
• The eigenspaces of $A$ are orthogonal to each other and have $n$ real eigenvalues counted with their algebraic multiplicities.