Eigenvalue for Orthogonal Matrix

[3.141592654]

Homework Statement

Let Q be an orthogonal matrix with an eigenvalue $λ_{1}$ = 1 and let x be an eigenvector belonging to $λ_{1}$. Show that x is also an eigenvector of $Q^{T}$.

Homework Equations

Qx = λx where x $\neq$ 0

The Attempt at a Solution

$Qx_{1} = x_{1}$ for some vector $x_{1}$

$(Qx_{1})^{T} = x_{1}^{T}Q^{T}$


I'm kind of stuck with how to start this problem, as I'm not sure what I've done is even starting down the right path. Can anyone give me a nudge in the right direction?

 

[Dick]

What's (Q^T)(Q) if Q is real orthogonal?

 

[3.141592654]

Thank you. I didn't realize what an orthogonal matrix was (yikes!). Once I did the proof fell right out of the definition.