Prove that all complex eigenvalues of the operators of a unitary or orthogonal representation have modulus 1.

Question

The question is given below:

Let $T$ be an orthogonal or unitary representation of the group $G$. Prove that all complex eigenvalues of the operators $T(g)$, $g \in G$ have modulus one.

But I do not know how the answer of it will differ from the answer given in this link:

Show that the eigenvalues of a unitary matrix have modulus $1$

And what are the relations between operators of orthogonal or unitary representation and unitary or orthogonal matrices?

Answer 1

An orthogonal representation is a representation by orthogonal matrices. A unitary representation is a representation by unitary matrices. So it really is just a question about eigenvalues of such matrices.

Answer 2

An orthogonal operator $A : X\rightarrow X$ on an inner product space $X$ satisfies $\|Ax\|=\|x\|$ for all $x$. If $A$ were to have an eigenvector $x\neq 0$ with eigenvalue $\lambda$, then $\|\lambda x\|=\|x\|$ or $|\lambda|\|x\|=\|x\|$ would have to hold, which would force $|\lambda|=1$.