Mixing
Prove $|tr(PAP') - tr(PBP')| leq ||A-B||_{infty} ||P'||_1^2$
$begingroup$
How can I show the following identity?
$|text{tr}(PAP') - text{tr}(PBP')| leq ||A-B||_{infty} ||P'||_1^2$
linear-algebra norm
asked Jan 25 at 0:37
shanishani
1458
$endgroup$
add a comment |
$begingroup$
How can I show the following identity?
$|text{tr}(PAP') - text{tr}(PBP')| leq ||A-B||_{infty} ||P'||_1^2$
linear-algebra norm
asked Jan 25 at 0:37
shanishani
1458
$endgroup$
3
$begingroup$
What is $P$ (and what happened to him)?
$endgroup$
– lcv
Jan 25 at 1:07
$begingroup$
@lcv: $P$ is a matrix that I am interested in estimating. I am not sure what you are trying to mean by "what happened to him"?
$endgroup$
– shani
Jan 25 at 1:20
$begingroup$
What is the relationship between $P$ and $P'$?
$endgroup$
– Kenny Wong
Jan 25 at 8:16
$begingroup$
$P'$ is the transpose of $P$.
$endgroup$
– shani
Jan 25 at 8:18
add a comment |
$begingroup$
How can I show the following identity?
$|text{tr}(PAP') - text{tr}(PBP')| leq ||A-B||_{infty} ||P'||_1^2$
linear-algebra norm
asked Jan 25 at 0:37
shanishani
1458
$endgroup$
How can I show the following identity?
$|text{tr}(PAP') - text{tr}(PBP')| leq ||A-B||_{infty} ||P'||_1^2$
linear-algebra norm
linear-algebra norm
asked Jan 25 at 0:37
shanishani
1458
asked Jan 25 at 0:37
shanishani
1458
asked Jan 25 at 0:37
shanishani
1458
asked Jan 25 at 0:37
shanishani
1458
asked Jan 25 at 0:37
shanishani
1458
1458
3
$begingroup$
What is $P$ (and what happened to him)?
$endgroup$
– lcv
Jan 25 at 1:07
$begingroup$
@lcv: $P$ is a matrix that I am interested in estimating. I am not sure what you are trying to mean by "what happened to him"?
$endgroup$
– shani
Jan 25 at 1:20
$begingroup$
What is the relationship between $P$ and $P'$?
$endgroup$
– Kenny Wong
Jan 25 at 8:16
$begingroup$
$P'$ is the transpose of $P$.
$endgroup$
– shani
Jan 25 at 8:18
add a comment |
3
$begingroup$
What is $P$ (and what happened to him)?
$endgroup$
– lcv
Jan 25 at 1:07
$begingroup$
@lcv: $P$ is a matrix that I am interested in estimating. I am not sure what you are trying to mean by "what happened to him"?
$endgroup$
– shani
Jan 25 at 1:20
$begingroup$
What is the relationship between $P$ and $P'$?
$endgroup$
– Kenny Wong
Jan 25 at 8:16
$begingroup$
$P'$ is the transpose of $P$.
$endgroup$
– shani
Jan 25 at 8:18
3
3
$begingroup$
What is $P$ (and what happened to him)?
$endgroup$
– lcv
Jan 25 at 1:07
$begingroup$
What is $P$ (and what happened to him)?
$endgroup$
– lcv
Jan 25 at 1:07
$begingroup$
@lcv: $P$ is a matrix that I am interested in estimating. I am not sure what you are trying to mean by "what happened to him"?
$endgroup$
– shani
Jan 25 at 1:20
$begingroup$
@lcv: $P$ is a matrix that I am interested in estimating. I am not sure what you are trying to mean by "what happened to him"?
$endgroup$
– shani
Jan 25 at 1:20
$begingroup$
What is the relationship between $P$ and $P'$?
$endgroup$
– Kenny Wong
Jan 25 at 8:16
$begingroup$
What is the relationship between $P$ and $P'$?
$endgroup$
– Kenny Wong
Jan 25 at 8:16
$begingroup$
$P'$ is the transpose of $P$.
$endgroup$
– shani
Jan 25 at 8:18
$begingroup$
$P'$ is the transpose of $P$.
$endgroup$
– shani
Jan 25 at 8:18
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You have
$$
|text{tr}(PAP') - text{tr}(PBP')|=|text{tr}(PAP'-PBP')|=|text{tr}(P(A-B)P')|=|text{tr}((A-B)P'P)|.
$$
Now you apply the well-known Hölder inequality
$$
|text{tr}(XY)|leq |X|_infty,|Y|_1.
$$
If you don't see it, here are more details. From the above, you get
$$
|text{tr}(PAP') - text{tr}(PBP')|leq|A-B|_infty,|P'P|_1.
$$
With ${mu_j}$ the singular values of $P$, you have $$|P'P|_1=sum_jmu_j^2leqleft(sum_jmu_jright)^2=|P|_1^2.$$
answered Jan 25 at 1:36
Martin ArgeramiMartin Argerami
128k1184184
$endgroup$
$begingroup$
Is there a type in the Holder's inequality? What happened to Y? I can't see how this can be used to prove the upper bound
$endgroup$
– shani
Jan 25 at 2:05
$begingroup$
Yes, the second one was a $Y$.
$endgroup$
– Martin Argerami
Jan 25 at 2:05
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You have
$$
|text{tr}(PAP') - text{tr}(PBP')|=|text{tr}(PAP'-PBP')|=|text{tr}(P(A-B)P')|=|text{tr}((A-B)P'P)|.
$$
Now you apply the well-known Hölder inequality
$$
|text{tr}(XY)|leq |X|_infty,|Y|_1.
$$
If you don't see it, here are more details. From the above, you get
$$
|text{tr}(PAP') - text{tr}(PBP')|leq|A-B|_infty,|P'P|_1.
$$
With ${mu_j}$ the singular values of $P$, you have $$|P'P|_1=sum_jmu_j^2leqleft(sum_jmu_jright)^2=|P|_1^2.$$
answered Jan 25 at 1:36
Martin ArgeramiMartin Argerami
128k1184184
$endgroup$
$begingroup$
Is there a type in the Holder's inequality? What happened to Y? I can't see how this can be used to prove the upper bound
$endgroup$
– shani
Jan 25 at 2:05
$begingroup$
Yes, the second one was a $Y$.
$endgroup$
– Martin Argerami
Jan 25 at 2:05
add a comment |
$begingroup$
You have
$$
|text{tr}(PAP') - text{tr}(PBP')|=|text{tr}(PAP'-PBP')|=|text{tr}(P(A-B)P')|=|text{tr}((A-B)P'P)|.
$$
Now you apply the well-known Hölder inequality
$$
|text{tr}(XY)|leq |X|_infty,|Y|_1.
$$
If you don't see it, here are more details. From the above, you get
$$
|text{tr}(PAP') - text{tr}(PBP')|leq|A-B|_infty,|P'P|_1.
$$
With ${mu_j}$ the singular values of $P$, you have $$|P'P|_1=sum_jmu_j^2leqleft(sum_jmu_jright)^2=|P|_1^2.$$
answered Jan 25 at 1:36
Martin ArgeramiMartin Argerami
128k1184184
$endgroup$
$begingroup$
Is there a type in the Holder's inequality? What happened to Y? I can't see how this can be used to prove the upper bound
$endgroup$
– shani
Jan 25 at 2:05
$begingroup$
Yes, the second one was a $Y$.
$endgroup$
– Martin Argerami
Jan 25 at 2:05
add a comment |
$begingroup$
You have
$$
|text{tr}(PAP') - text{tr}(PBP')|=|text{tr}(PAP'-PBP')|=|text{tr}(P(A-B)P')|=|text{tr}((A-B)P'P)|.
$$
Now you apply the well-known Hölder inequality
$$
|text{tr}(XY)|leq |X|_infty,|Y|_1.
$$
If you don't see it, here are more details. From the above, you get
$$
|text{tr}(PAP') - text{tr}(PBP')|leq|A-B|_infty,|P'P|_1.
$$
With ${mu_j}$ the singular values of $P$, you have $$|P'P|_1=sum_jmu_j^2leqleft(sum_jmu_jright)^2=|P|_1^2.$$
answered Jan 25 at 1:36
Martin ArgeramiMartin Argerami
128k1184184
$endgroup$
You have
$$
|text{tr}(PAP') - text{tr}(PBP')|=|text{tr}(PAP'-PBP')|=|text{tr}(P(A-B)P')|=|text{tr}((A-B)P'P)|.
$$
Now you apply the well-known Hölder inequality
$$
|text{tr}(XY)|leq |X|_infty,|Y|_1.
$$
If you don't see it, here are more details. From the above, you get
$$
|text{tr}(PAP') - text{tr}(PBP')|leq|A-B|_infty,|P'P|_1.
$$
With ${mu_j}$ the singular values of $P$, you have $$|P'P|_1=sum_jmu_j^2leqleft(sum_jmu_jright)^2=|P|_1^2.$$
answered Jan 25 at 1:36
Martin ArgeramiMartin Argerami
128k1184184
edited Jan 25 at 3:52
answered Jan 25 at 1:36
Martin ArgeramiMartin Argerami
128k1184184
answered Jan 25 at 1:36
Martin ArgeramiMartin Argerami
128k1184184
answered Jan 25 at 1:36
Martin ArgeramiMartin Argerami
128k1184184
128k1184184
$begingroup$
Is there a type in the Holder's inequality? What happened to Y? I can't see how this can be used to prove the upper bound
$endgroup$
– shani
Jan 25 at 2:05
$begingroup$
Yes, the second one was a $Y$.
$endgroup$
– Martin Argerami
Jan 25 at 2:05
add a comment |
$begingroup$
Is there a type in the Holder's inequality? What happened to Y? I can't see how this can be used to prove the upper bound
$endgroup$
– shani
Jan 25 at 2:05
$begingroup$
Yes, the second one was a $Y$.
$endgroup$
– Martin Argerami
Jan 25 at 2:05
$begingroup$
Is there a type in the Holder's inequality? What happened to Y? I can't see how this can be used to prove the upper bound
$endgroup$
– shani
Jan 25 at 2:05
$begingroup$
Is there a type in the Holder's inequality? What happened to Y? I can't see how this can be used to prove the upper bound
$endgroup$
– shani
Jan 25 at 2:05
$begingroup$
Yes, the second one was a $Y$.
$endgroup$
– Martin Argerami
Jan 25 at 2:05
$begingroup$
Yes, the second one was a $Y$.
$endgroup$
– Martin Argerami
Jan 25 at 2:05
add a comment |
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3
$begingroup$
What is $P$ (and what happened to him)?
$endgroup$
– lcv
Jan 25 at 1:07
$begingroup$
@lcv: $P$ is a matrix that I am interested in estimating. I am not sure what you are trying to mean by "what happened to him"?
$endgroup$
– shani
Jan 25 at 1:20
$begingroup$
What is the relationship between $P$ and $P'$?
$endgroup$
– Kenny Wong
Jan 25 at 8:16
$begingroup$
$P'$ is the transpose of $P$.
$endgroup$
– shani
Jan 25 at 8:18