Mixing
Prove that all complex eigenvalues of the operators of a unitary or orthogonal representation have modulus 1.
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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?
linear-algebra functional-analysis operator-theory representation-theory orthogonal-matrices
edited Feb 1 at 1:41
Theo Bendit
20.8k12354
asked Feb 1 at 1:26
hopefullyhopefully
281215
$endgroup$
add a comment |
$begingroup$
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?
linear-algebra functional-analysis operator-theory representation-theory orthogonal-matrices
edited Feb 1 at 1:41
Theo Bendit
20.8k12354
asked Feb 1 at 1:26
hopefullyhopefully
281215
$endgroup$
$begingroup$
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.
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– Theo Bendit
Feb 1 at 1:43
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Okay I am sorry @TheoBendit I will obey this rule as much as possible.
$endgroup$
– hopefully
Feb 1 at 1:55
$begingroup$
No worries. Images are primarily used to embed diagrams, or other things that might aid understanding that cannot be formatted.
$endgroup$
– Theo Bendit
Feb 1 at 1:57
add a comment |
$begingroup$
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?
linear-algebra functional-analysis operator-theory representation-theory orthogonal-matrices
edited Feb 1 at 1:41
Theo Bendit
20.8k12354
asked Feb 1 at 1:26
hopefullyhopefully
281215
$endgroup$
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?
linear-algebra functional-analysis operator-theory representation-theory orthogonal-matrices
linear-algebra functional-analysis operator-theory representation-theory orthogonal-matrices
edited Feb 1 at 1:41
Theo Bendit
20.8k12354
asked Feb 1 at 1:26
hopefullyhopefully
281215
edited Feb 1 at 1:41
Theo Bendit
20.8k12354
asked Feb 1 at 1:26
hopefullyhopefully
281215
edited Feb 1 at 1:41
Theo Bendit
20.8k12354
edited Feb 1 at 1:41
Theo Bendit
20.8k12354
edited Feb 1 at 1:41
Theo Bendit
20.8k12354
20.8k12354
asked Feb 1 at 1:26
hopefullyhopefully
281215
asked Feb 1 at 1:26
hopefullyhopefully
281215
asked Feb 1 at 1:26
hopefullyhopefully
281215
281215
$begingroup$
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.
$endgroup$
– Theo Bendit
Feb 1 at 1:43
$begingroup$
Okay I am sorry @TheoBendit I will obey this rule as much as possible.
$endgroup$
– hopefully
Feb 1 at 1:55
$begingroup$
No worries. Images are primarily used to embed diagrams, or other things that might aid understanding that cannot be formatted.
$endgroup$
– Theo Bendit
Feb 1 at 1:57
add a comment |
$begingroup$
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.
$endgroup$
– Theo Bendit
Feb 1 at 1:43
$begingroup$
Okay I am sorry @TheoBendit I will obey this rule as much as possible.
$endgroup$
– hopefully
Feb 1 at 1:55
$begingroup$
No worries. Images are primarily used to embed diagrams, or other things that might aid understanding that cannot be formatted.
$endgroup$
– Theo Bendit
Feb 1 at 1:57
$begingroup$
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.
$endgroup$
– Theo Bendit
Feb 1 at 1:43
$begingroup$
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.
$endgroup$
– Theo Bendit
Feb 1 at 1:43
$begingroup$
Okay I am sorry @TheoBendit I will obey this rule as much as possible.
$endgroup$
– hopefully
Feb 1 at 1:55
$begingroup$
Okay I am sorry @TheoBendit I will obey this rule as much as possible.
$endgroup$
– hopefully
Feb 1 at 1:55
$begingroup$
No worries. Images are primarily used to embed diagrams, or other things that might aid understanding that cannot be formatted.
$endgroup$
– Theo Bendit
Feb 1 at 1:57
$begingroup$
No worries. Images are primarily used to embed diagrams, or other things that might aid understanding that cannot be formatted.
$endgroup$
– Theo Bendit
Feb 1 at 1:57
add a comment |
2 Answers
2
active
oldest
votes
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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.
answered Feb 3 at 2:52
Gerry MyersonGerry Myerson
148k8152306
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$begingroup$
So you mean that the answer to the above question is the same as the answer in the link mentioned above?
$endgroup$
– hopefully
Feb 3 at 2:56
1
$begingroup$
Well, the question is the same as the one in the link. I haven't visited the link to see what the answer there says.
$endgroup$
– Gerry Myerson
Feb 3 at 3:02
add a comment |
$begingroup$
An orthogonal operator $A : Xrightarrow X$ on an inner product space $X$ satisfies $|Ax|=|x|$ for all $x$. If $A$ were to have an eigenvector $xneq 0$ with eigenvalue $lambda$, then $|lambda x|=|x|$ or $|lambda||x|=|x|$ would have to hold, which would force $|lambda|=1$.
answered Feb 3 at 4:25
DisintegratingByPartsDisintegratingByParts
60.3k42681
$endgroup$
$begingroup$
So you are saying that the above link does not contain the answer?
$endgroup$
– hopefully
Feb 4 at 0:15
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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.
answered Feb 3 at 2:52
Gerry MyersonGerry Myerson
148k8152306
$endgroup$
$begingroup$
So you mean that the answer to the above question is the same as the answer in the link mentioned above?
$endgroup$
– hopefully
Feb 3 at 2:56
1
$begingroup$
Well, the question is the same as the one in the link. I haven't visited the link to see what the answer there says.
$endgroup$
– Gerry Myerson
Feb 3 at 3:02
add a comment |
$begingroup$
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.
answered Feb 3 at 2:52
Gerry MyersonGerry Myerson
148k8152306
$endgroup$
$begingroup$
So you mean that the answer to the above question is the same as the answer in the link mentioned above?
$endgroup$
– hopefully
Feb 3 at 2:56
1
$begingroup$
Well, the question is the same as the one in the link. I haven't visited the link to see what the answer there says.
$endgroup$
– Gerry Myerson
Feb 3 at 3:02
add a comment |
$begingroup$
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.
answered Feb 3 at 2:52
Gerry MyersonGerry Myerson
148k8152306
$endgroup$
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.
answered Feb 3 at 2:52
Gerry MyersonGerry Myerson
148k8152306
answered Feb 3 at 2:52
Gerry MyersonGerry Myerson
148k8152306
answered Feb 3 at 2:52
Gerry MyersonGerry Myerson
148k8152306
answered Feb 3 at 2:52
Gerry MyersonGerry Myerson
148k8152306
148k8152306
$begingroup$
So you mean that the answer to the above question is the same as the answer in the link mentioned above?
$endgroup$
– hopefully
Feb 3 at 2:56
1
$begingroup$
Well, the question is the same as the one in the link. I haven't visited the link to see what the answer there says.
$endgroup$
– Gerry Myerson
Feb 3 at 3:02
add a comment |
$begingroup$
So you mean that the answer to the above question is the same as the answer in the link mentioned above?
$endgroup$
– hopefully
Feb 3 at 2:56
1
$begingroup$
Well, the question is the same as the one in the link. I haven't visited the link to see what the answer there says.
$endgroup$
– Gerry Myerson
Feb 3 at 3:02
$begingroup$
So you mean that the answer to the above question is the same as the answer in the link mentioned above?
$endgroup$
– hopefully
Feb 3 at 2:56
$begingroup$
So you mean that the answer to the above question is the same as the answer in the link mentioned above?
$endgroup$
– hopefully
Feb 3 at 2:56
1
1
$begingroup$
Well, the question is the same as the one in the link. I haven't visited the link to see what the answer there says.
$endgroup$
– Gerry Myerson
Feb 3 at 3:02
$begingroup$
Well, the question is the same as the one in the link. I haven't visited the link to see what the answer there says.
$endgroup$
– Gerry Myerson
Feb 3 at 3:02
add a comment |
$begingroup$
An orthogonal operator $A : Xrightarrow X$ on an inner product space $X$ satisfies $|Ax|=|x|$ for all $x$. If $A$ were to have an eigenvector $xneq 0$ with eigenvalue $lambda$, then $|lambda x|=|x|$ or $|lambda||x|=|x|$ would have to hold, which would force $|lambda|=1$.
answered Feb 3 at 4:25
DisintegratingByPartsDisintegratingByParts
60.3k42681
$endgroup$
$begingroup$
So you are saying that the above link does not contain the answer?
$endgroup$
– hopefully
Feb 4 at 0:15
add a comment |
$begingroup$
An orthogonal operator $A : Xrightarrow X$ on an inner product space $X$ satisfies $|Ax|=|x|$ for all $x$. If $A$ were to have an eigenvector $xneq 0$ with eigenvalue $lambda$, then $|lambda x|=|x|$ or $|lambda||x|=|x|$ would have to hold, which would force $|lambda|=1$.
answered Feb 3 at 4:25
DisintegratingByPartsDisintegratingByParts
60.3k42681
$endgroup$
$begingroup$
So you are saying that the above link does not contain the answer?
$endgroup$
– hopefully
Feb 4 at 0:15
add a comment |
$begingroup$
An orthogonal operator $A : Xrightarrow X$ on an inner product space $X$ satisfies $|Ax|=|x|$ for all $x$. If $A$ were to have an eigenvector $xneq 0$ with eigenvalue $lambda$, then $|lambda x|=|x|$ or $|lambda||x|=|x|$ would have to hold, which would force $|lambda|=1$.
answered Feb 3 at 4:25
DisintegratingByPartsDisintegratingByParts
60.3k42681
$endgroup$
An orthogonal operator $A : Xrightarrow X$ on an inner product space $X$ satisfies $|Ax|=|x|$ for all $x$. If $A$ were to have an eigenvector $xneq 0$ with eigenvalue $lambda$, then $|lambda x|=|x|$ or $|lambda||x|=|x|$ would have to hold, which would force $|lambda|=1$.
answered Feb 3 at 4:25
DisintegratingByPartsDisintegratingByParts
60.3k42681
answered Feb 3 at 4:25
DisintegratingByPartsDisintegratingByParts
60.3k42681
answered Feb 3 at 4:25
DisintegratingByPartsDisintegratingByParts
60.3k42681
answered Feb 3 at 4:25
DisintegratingByPartsDisintegratingByParts
60.3k42681
60.3k42681
$begingroup$
So you are saying that the above link does not contain the answer?
$endgroup$
– hopefully
Feb 4 at 0:15
add a comment |
$begingroup$
So you are saying that the above link does not contain the answer?
$endgroup$
– hopefully
Feb 4 at 0:15
$begingroup$
So you are saying that the above link does not contain the answer?
$endgroup$
– hopefully
Feb 4 at 0:15
$begingroup$
So you are saying that the above link does not contain the answer?
$endgroup$
– hopefully
Feb 4 at 0:15
add a comment |
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$begingroup$
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.
$endgroup$
– Theo Bendit
Feb 1 at 1:43
$begingroup$
Okay I am sorry @TheoBendit I will obey this rule as much as possible.
$endgroup$
– hopefully
Feb 1 at 1:55
$begingroup$
No worries. Images are primarily used to embed diagrams, or other things that might aid understanding that cannot be formatted.
$endgroup$
– Theo Bendit
Feb 1 at 1:57