TITLE: Are all eigenvectors, of any matrix, always orthogonal?
QUESTION [64 upvotes]: I have a very simple question that can be stated without proof. Are all eigenvectors, of any matrix, always orthogonal? I am trying to understand Principal components and it is cruucial for me to see the basis of eigenvectors.

REPLY [2 votes]: Not necessarily all orthogonal. However two eigenvectors corresponding to different eigenvalues are orthogonal.
e.g Let  $X_1$ and $X_2$ be two eigenvectors of a matrix $A$ corresponding to eigenvalues $\lambda_1 $ and $\lambda_2$ where $\lambda_1 \ne \lambda_2$.
Now,
$AX_1 = \lambda_1 X_1$ and  $AX_2 = \lambda_2 X_2$.
Taking Transpose of first,
$
\begin{align}
&(AX_1)^T = (\lambda_1 X_1)^T\\
\implies & X_1^TA^T = \lambda_1 X_1^T\\
\implies &X_1^TA^TX_2 = \lambda_1 X_1^T X_2\\
\implies & X_1^T\lambda_2 X_2 = \lambda_1 X_1^T X_2\\
\implies &\lambda_2X_1^T X_2 = \lambda_1 X_1^T X_2\\
\implies &(\lambda_2 - \lambda_1)X_1^T X_2 = 0\\
\end{align}
$
Since $\lambda_2 \ne \lambda_1$, $X_1^T X_2 $ must be $0$. This establishes orthogonality of eigenvectors corresponding to two different eigenvalues.