$S=S^T\to X\Lambda X^{-1}=(X^{-1})^T\Lambda X^T\to X^TX=I$

That makes each eigenvector in $X$ orthogonal to the other eigenvectors when $S=S^T$

• A symmetric matrix has only real eigenvalues
• The eigenvectors can be chosen orthonormal.

Spectral Theorem

Every symmetric matrix has the factorization $S = Q\Lambda Q^T$ with real eigenvalues in $\Lambda$ and orthonormal eigenvectors in the columns of $Q$:

Symmetric diagonalization $S=Q\Lambda Q^{-1}=Q\Lambda Q^T$

Remember in this case row space and column space are the same.

Real Eigenvalues

All the eigenvalues of a real symmetric matrix are real.

Proof

Orthogonal Eigenvectors

Eigenvectors of a real symmetric matrix(when they correspond to different X's) are always perpendicular.

Proof

Every symmetric matrix is a combination of perpendicular projection matrixes.

Complex Eigenvalues of Real Matrices