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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?

Theo Bendit
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Intuition
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  • That's weird: from your previous questions, it's clear that you can use MathJax, yet you still embedded the question in an image? Please format all questions in text and MathJax as much as possible, to help people search for your question. – Theo Bendit Feb 01 '19 at 01:43
  • Okay I am sorry @TheoBendit I will obey this rule as much as possible. – Intuition Feb 01 '19 at 01:55
  • No worries. Images are primarily used to embed diagrams, or other things that might aid understanding that cannot be formatted. – Theo Bendit Feb 01 '19 at 01:57

2 Answers2

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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.

Gerry Myerson
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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 $|\lambda|=1$.

Disintegrating By Parts
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