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.