# Math Help - Orthogonal Matrix

1. ## Orthogonal Matrix

Let $\alpha$ be an imaginary eigenvalue of an orthogonal matrix Q adn let X be a corresponding eigenvector. Prove that $X^T X = 0$

Ok here is what I did:

$X^TQ^{-1}QX = X^T Q^{-1} \alpha X = \alpha X^T Q^{-1}X = \alpha X^T Q^T X = \alpha (QX)^T X$

$= \alpha ( \alpha X)^T X = a X \overline{\alpha} X = \alpha \overline{\alpha} X^T X = {|\alpha|}^2 X^T X$

Also $X^TQ^{-1}QX = X^T X$

So $X^T X = {|\alpha|}^2 X^T X$
i.e. $(1-{|\alpha|}^2) X^T X = 0$

i.e. $X^T X = 0$ or $(1- {|\alpha|}^2) = 0$

Why can't $(1- {|\alpha|}^2) = 0$ ?

2. Hi, it should be alpha^2 and not |alpha|^2 cause you don't take complex conjugate but rather the transpose only, and because alpha=ai a is real, alpha^2=-a^2 so it cannot be equal 1.