Differentiability of operator norm [closed]A matrix inequality involving the Hilbert-Schmidt normAn inequality involving operator and trace normsNorm of a matrix operator with a special structureAlmost commuting unitary matricesNorm of an operator formed using a unitary operatorWhat is the intuition for the trace norm (nuclear norm)?What's the best orthonormal matrix to align two matrices in the operator norm sense?Reference Request: Differentiability of Moreau EnvelopeQuaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiabilityOperator norm of a soft thresholded symmetric matrix
Differentiability of operator norm [closed]
A matrix inequality involving the Hilbert-Schmidt normAn inequality involving operator and trace normsNorm of a matrix operator with a special structureAlmost commuting unitary matricesNorm of an operator formed using a unitary operatorWhat is the intuition for the trace norm (nuclear norm)?What's the best orthonormal matrix to align two matrices in the operator norm sense?Reference Request: Differentiability of Moreau EnvelopeQuaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiabilityOperator norm of a soft thresholded symmetric matrix
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Is there any known results about differentiability properties of the function $mathbb f:mathbb R tomathbb R,$ $f(t):=|A+tB|_op$ where $|.|_op$ denotes the usual operator norm of the matrices acting on finite dimensional complex Hilbert spaces?
matrices matrix-analysis differential-operators matrix-theory
asked Aug 7 at 12:19
A beginner mathmaticianA beginner mathmatician
3246 bronze badges
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closed as too broad by Deane Yang, Yemon Choi, Pace Nielsen, Joonas Ilmavirta, Ben McKay Aug 13 at 9:56
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
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Is there any known results about differentiability properties of the function $mathbb f:mathbb R tomathbb R,$ $f(t):=|A+tB|_op$ where $|.|_op$ denotes the usual operator norm of the matrices acting on finite dimensional complex Hilbert spaces?
matrices matrix-analysis differential-operators matrix-theory
asked Aug 7 at 12:19
A beginner mathmaticianA beginner mathmatician
3246 bronze badges
$endgroup$
closed as too broad by Deane Yang, Yemon Choi, Pace Nielsen, Joonas Ilmavirta, Ben McKay Aug 13 at 9:56
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
3
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note that the operator norm, restricted to diagonal matrices, is just the max norm.
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– Pietro Majer
Aug 7 at 14:18
1
$begingroup$
Maybe this question should be migrated to math.stackexchange.con.
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– Deane Yang
Aug 8 at 0:30
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@DeaneYang I don't think it needs to go to MSE because of the level of the question, but I do think that in its current form it is too broad and amounts to the OP asking for a lesson or a wikipedia entry, rather than the answer to a particular question
$endgroup$
– Yemon Choi
Aug 10 at 3:41
add a comment |
$begingroup$
Is there any known results about differentiability properties of the function $mathbb f:mathbb R tomathbb R,$ $f(t):=|A+tB|_op$ where $|.|_op$ denotes the usual operator norm of the matrices acting on finite dimensional complex Hilbert spaces?
matrices matrix-analysis differential-operators matrix-theory
asked Aug 7 at 12:19
A beginner mathmaticianA beginner mathmatician
3246 bronze badges
$endgroup$
Is there any known results about differentiability properties of the function $mathbb f:mathbb R tomathbb R,$ $f(t):=|A+tB|_op$ where $|.|_op$ denotes the usual operator norm of the matrices acting on finite dimensional complex Hilbert spaces?
matrices matrix-analysis differential-operators matrix-theory
matrices matrix-analysis differential-operators matrix-theory
asked Aug 7 at 12:19
A beginner mathmaticianA beginner mathmatician
3246 bronze badges
asked Aug 7 at 12:19
A beginner mathmaticianA beginner mathmatician
3246 bronze badges
asked Aug 7 at 12:19
A beginner mathmaticianA beginner mathmatician
3246 bronze badges
asked Aug 7 at 12:19
A beginner mathmaticianA beginner mathmatician
3246 bronze badges
asked Aug 7 at 12:19
A beginner mathmaticianA beginner mathmatician
3246 bronze badges
3246 bronze badges
closed as too broad by Deane Yang, Yemon Choi, Pace Nielsen, Joonas Ilmavirta, Ben McKay Aug 13 at 9:56
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as too broad by Deane Yang, Yemon Choi, Pace Nielsen, Joonas Ilmavirta, Ben McKay Aug 13 at 9:56
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as too broad by Deane Yang, Yemon Choi, Pace Nielsen, Joonas Ilmavirta, Ben McKay Aug 13 at 9:56
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
3
$begingroup$
note that the operator norm, restricted to diagonal matrices, is just the max norm.
$endgroup$
– Pietro Majer
Aug 7 at 14:18
1
$begingroup$
Maybe this question should be migrated to math.stackexchange.con.
$endgroup$
– Deane Yang
Aug 8 at 0:30
$begingroup$
@DeaneYang I don't think it needs to go to MSE because of the level of the question, but I do think that in its current form it is too broad and amounts to the OP asking for a lesson or a wikipedia entry, rather than the answer to a particular question
$endgroup$
– Yemon Choi
Aug 10 at 3:41
add a comment |
3
$begingroup$
note that the operator norm, restricted to diagonal matrices, is just the max norm.
$endgroup$
– Pietro Majer
Aug 7 at 14:18
1
$begingroup$
Maybe this question should be migrated to math.stackexchange.con.
$endgroup$
– Deane Yang
Aug 8 at 0:30
$begingroup$
@DeaneYang I don't think it needs to go to MSE because of the level of the question, but I do think that in its current form it is too broad and amounts to the OP asking for a lesson or a wikipedia entry, rather than the answer to a particular question
$endgroup$
– Yemon Choi
Aug 10 at 3:41
3
3
$begingroup$
note that the operator norm, restricted to diagonal matrices, is just the max norm.
$endgroup$
– Pietro Majer
Aug 7 at 14:18
$begingroup$
note that the operator norm, restricted to diagonal matrices, is just the max norm.
$endgroup$
– Pietro Majer
Aug 7 at 14:18
1
1
$begingroup$
Maybe this question should be migrated to math.stackexchange.con.
$endgroup$
– Deane Yang
Aug 8 at 0:30
$begingroup$
Maybe this question should be migrated to math.stackexchange.con.
$endgroup$
– Deane Yang
Aug 8 at 0:30
$begingroup$
@DeaneYang I don't think it needs to go to MSE because of the level of the question, but I do think that in its current form it is too broad and amounts to the OP asking for a lesson or a wikipedia entry, rather than the answer to a particular question
$endgroup$
– Yemon Choi
Aug 10 at 3:41
$begingroup$
@DeaneYang I don't think it needs to go to MSE because of the level of the question, but I do think that in its current form it is too broad and amounts to the OP asking for a lesson or a wikipedia entry, rather than the answer to a particular question
$endgroup$
– Yemon Choi
Aug 10 at 3:41
add a comment |
1 Answer
1
active
oldest
votes
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It need not be differentiable everywhere. Let $P$ and $Q$ be mutually orthogonal self-adjoint projections. Then the norm of $P+ t Q$ is 1 for $|t| leq 1$ and $|t|$ for $|t| > 1$.
However, $|A + tB|$ is Lipschitz in $t$, so it is differentiable almost everywhere by Rademacher's theorem.
answered Aug 7 at 13:14
Nik WeaverNik Weaver
24k1 gold badge52 silver badges140 bronze badges
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Yes. I can see that. But what I meant is that do we know some nontrivial things, e.g. if it is k times differentiable or not etc.
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– Samya Ray
Aug 7 at 21:10
3
$begingroup$
Or $A=0$, $B=1$, so $f(t)=|t|$.
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– Christian Remling
Aug 7 at 22:06
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@SamyaRay It is not fair to ask for people to write down "some nontrivial things" because you are essentially asking other people to do the work of figuring out what questions you have.
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– Yemon Choi
Aug 10 at 3:40
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Moreover, why do you have two separate accounts? It would be best if you ask the moderators to merge them
$endgroup$
– Yemon Choi
Aug 10 at 3:42
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Okay. I understand. I wrongly commented at he place where I should have not.
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– A beginner mathmatician
Aug 10 at 4:28
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It need not be differentiable everywhere. Let $P$ and $Q$ be mutually orthogonal self-adjoint projections. Then the norm of $P+ t Q$ is 1 for $|t| leq 1$ and $|t|$ for $|t| > 1$.
However, $|A + tB|$ is Lipschitz in $t$, so it is differentiable almost everywhere by Rademacher's theorem.
answered Aug 7 at 13:14
Nik WeaverNik Weaver
24k1 gold badge52 silver badges140 bronze badges
$endgroup$
$begingroup$
Yes. I can see that. But what I meant is that do we know some nontrivial things, e.g. if it is k times differentiable or not etc.
$endgroup$
– Samya Ray
Aug 7 at 21:10
3
$begingroup$
Or $A=0$, $B=1$, so $f(t)=|t|$.
$endgroup$
– Christian Remling
Aug 7 at 22:06
$begingroup$
@SamyaRay It is not fair to ask for people to write down "some nontrivial things" because you are essentially asking other people to do the work of figuring out what questions you have.
$endgroup$
– Yemon Choi
Aug 10 at 3:40
$begingroup$
Moreover, why do you have two separate accounts? It would be best if you ask the moderators to merge them
$endgroup$
– Yemon Choi
Aug 10 at 3:42
$begingroup$
Okay. I understand. I wrongly commented at he place where I should have not.
$endgroup$
– A beginner mathmatician
Aug 10 at 4:28
add a comment |
$begingroup$
It need not be differentiable everywhere. Let $P$ and $Q$ be mutually orthogonal self-adjoint projections. Then the norm of $P+ t Q$ is 1 for $|t| leq 1$ and $|t|$ for $|t| > 1$.
However, $|A + tB|$ is Lipschitz in $t$, so it is differentiable almost everywhere by Rademacher's theorem.
answered Aug 7 at 13:14
Nik WeaverNik Weaver
24k1 gold badge52 silver badges140 bronze badges
$endgroup$
$begingroup$
Yes. I can see that. But what I meant is that do we know some nontrivial things, e.g. if it is k times differentiable or not etc.
$endgroup$
– Samya Ray
Aug 7 at 21:10
3
$begingroup$
Or $A=0$, $B=1$, so $f(t)=|t|$.
$endgroup$
– Christian Remling
Aug 7 at 22:06
$begingroup$
@SamyaRay It is not fair to ask for people to write down "some nontrivial things" because you are essentially asking other people to do the work of figuring out what questions you have.
$endgroup$
– Yemon Choi
Aug 10 at 3:40
$begingroup$
Moreover, why do you have two separate accounts? It would be best if you ask the moderators to merge them
$endgroup$
– Yemon Choi
Aug 10 at 3:42
$begingroup$
Okay. I understand. I wrongly commented at he place where I should have not.
$endgroup$
– A beginner mathmatician
Aug 10 at 4:28
add a comment |
$begingroup$
It need not be differentiable everywhere. Let $P$ and $Q$ be mutually orthogonal self-adjoint projections. Then the norm of $P+ t Q$ is 1 for $|t| leq 1$ and $|t|$ for $|t| > 1$.
However, $|A + tB|$ is Lipschitz in $t$, so it is differentiable almost everywhere by Rademacher's theorem.
answered Aug 7 at 13:14
Nik WeaverNik Weaver
24k1 gold badge52 silver badges140 bronze badges
$endgroup$
It need not be differentiable everywhere. Let $P$ and $Q$ be mutually orthogonal self-adjoint projections. Then the norm of $P+ t Q$ is 1 for $|t| leq 1$ and $|t|$ for $|t| > 1$.
However, $|A + tB|$ is Lipschitz in $t$, so it is differentiable almost everywhere by Rademacher's theorem.
answered Aug 7 at 13:14
Nik WeaverNik Weaver
24k1 gold badge52 silver badges140 bronze badges
edited Aug 7 at 13:32
answered Aug 7 at 13:14
Nik WeaverNik Weaver
24k1 gold badge52 silver badges140 bronze badges
answered Aug 7 at 13:14
Nik WeaverNik Weaver
24k1 gold badge52 silver badges140 bronze badges
answered Aug 7 at 13:14
Nik WeaverNik Weaver
24k1 gold badge52 silver badges140 bronze badges
24k1 gold badge52 silver badges140 bronze badges
$begingroup$
Yes. I can see that. But what I meant is that do we know some nontrivial things, e.g. if it is k times differentiable or not etc.
$endgroup$
– Samya Ray
Aug 7 at 21:10
3
$begingroup$
Or $A=0$, $B=1$, so $f(t)=|t|$.
$endgroup$
– Christian Remling
Aug 7 at 22:06
$begingroup$
@SamyaRay It is not fair to ask for people to write down "some nontrivial things" because you are essentially asking other people to do the work of figuring out what questions you have.
$endgroup$
– Yemon Choi
Aug 10 at 3:40
$begingroup$
Moreover, why do you have two separate accounts? It would be best if you ask the moderators to merge them
$endgroup$
– Yemon Choi
Aug 10 at 3:42
$begingroup$
Okay. I understand. I wrongly commented at he place where I should have not.
$endgroup$
– A beginner mathmatician
Aug 10 at 4:28
add a comment |
$begingroup$
Yes. I can see that. But what I meant is that do we know some nontrivial things, e.g. if it is k times differentiable or not etc.
$endgroup$
– Samya Ray
Aug 7 at 21:10
3
$begingroup$
Or $A=0$, $B=1$, so $f(t)=|t|$.
$endgroup$
– Christian Remling
Aug 7 at 22:06
$begingroup$
@SamyaRay It is not fair to ask for people to write down "some nontrivial things" because you are essentially asking other people to do the work of figuring out what questions you have.
$endgroup$
– Yemon Choi
Aug 10 at 3:40
$begingroup$
Moreover, why do you have two separate accounts? It would be best if you ask the moderators to merge them
$endgroup$
– Yemon Choi
Aug 10 at 3:42
$begingroup$
Okay. I understand. I wrongly commented at he place where I should have not.
$endgroup$
– A beginner mathmatician
Aug 10 at 4:28
$begingroup$
Yes. I can see that. But what I meant is that do we know some nontrivial things, e.g. if it is k times differentiable or not etc.
$endgroup$
– Samya Ray
Aug 7 at 21:10
$begingroup$
Yes. I can see that. But what I meant is that do we know some nontrivial things, e.g. if it is k times differentiable or not etc.
$endgroup$
– Samya Ray
Aug 7 at 21:10
3
3
$begingroup$
Or $A=0$, $B=1$, so $f(t)=|t|$.
$endgroup$
– Christian Remling
Aug 7 at 22:06
$begingroup$
Or $A=0$, $B=1$, so $f(t)=|t|$.
$endgroup$
– Christian Remling
Aug 7 at 22:06
$begingroup$
@SamyaRay It is not fair to ask for people to write down "some nontrivial things" because you are essentially asking other people to do the work of figuring out what questions you have.
$endgroup$
– Yemon Choi
Aug 10 at 3:40
$begingroup$
@SamyaRay It is not fair to ask for people to write down "some nontrivial things" because you are essentially asking other people to do the work of figuring out what questions you have.
$endgroup$
– Yemon Choi
Aug 10 at 3:40
$begingroup$
Moreover, why do you have two separate accounts? It would be best if you ask the moderators to merge them
$endgroup$
– Yemon Choi
Aug 10 at 3:42
$begingroup$
Moreover, why do you have two separate accounts? It would be best if you ask the moderators to merge them
$endgroup$
– Yemon Choi
Aug 10 at 3:42
$begingroup$
Okay. I understand. I wrongly commented at he place where I should have not.
$endgroup$
– A beginner mathmatician
Aug 10 at 4:28
$begingroup$
Okay. I understand. I wrongly commented at he place where I should have not.
$endgroup$
– A beginner mathmatician
Aug 10 at 4:28
add a comment |
3
$begingroup$
note that the operator norm, restricted to diagonal matrices, is just the max norm.
$endgroup$
– Pietro Majer
Aug 7 at 14:18
1
$begingroup$
Maybe this question should be migrated to math.stackexchange.con.
$endgroup$
– Deane Yang
Aug 8 at 0:30
$begingroup$
@DeaneYang I don't think it needs to go to MSE because of the level of the question, but I do think that in its current form it is too broad and amounts to the OP asking for a lesson or a wikipedia entry, rather than the answer to a particular question
$endgroup$
– Yemon Choi
Aug 10 at 3:41