Eigenvectors for Different Eigenvalues are Orthogonal for Symmetric Matrixes

Theorem

• If $A$ is a Symmetric Matrix (could be Unitary, Normal, etc..)
• With eigenvalues ${\lambda }_1,{\lambda }_2$ that are unique
• For eigenvector $x_1$ for ${\lambda }_1$, eigenvector $x_2$ for ${\lambda }_2$
• Then, $⟨x_1|x_2⟩=0$

Proof

1. Given $A{x}_1={\lambda }_1{x}_1$
2. Given $A{x}_2={\lambda }_2{x}_2$
3. $⟹(A{x}_1{)}^T=({\lambda }_1{x}_1{)}^T$
4. $⟹x_1^T{A}^T={\lambda }_1{x}_1^T$
5. $⟹x_1^T{A}^T{x}_2={\lambda }_1{x}_1^T{x}_2$
6. $⟹x_1^T{\lambda }_2{x}_2={\lambda }_1{x}_1^T{x}_2$
7. $⟹{\lambda }_2{x}_1^T{x}_2={\lambda }_1{x}_1^T{x}_2$
8. $⟹({\lambda }_2-{\lambda }_1)(x_1^T{x}_2)=0$
9. Note that since ${\lambda }_2\ne {\lambda }_1$, then it has to be that $x_1^T{x}_2=0$, which means $⟨x_1,x_2⟩=0$ by definition of Dot Product