How to deal with this 2-norm? [closed]
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$A$ is an $m×n$ matrix, verify the inequality $|A|_inftyleqsqrt n|A|_2$ and give an example of a nonzero matrix if the equality is achieved.
linear-algebra
asked Dec 18 '18 at 0:41
KristyKristy
454
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closed as off-topic by Saad, Leucippus, Eevee Trainer, A Blumenthal, mrtaurho Dec 18 '18 at 5:39
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Leucippus, Eevee Trainer, A Blumenthal, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
$A$ is an $m×n$ matrix, verify the inequality $|A|_inftyleqsqrt n|A|_2$ and give an example of a nonzero matrix if the equality is achieved.
linear-algebra
asked Dec 18 '18 at 0:41
KristyKristy
454
$endgroup$
closed as off-topic by Saad, Leucippus, Eevee Trainer, A Blumenthal, mrtaurho Dec 18 '18 at 5:39
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Leucippus, Eevee Trainer, A Blumenthal, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
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Where are you stuck? Maybe start with reviewing the definitions of these norms.
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– Dave
Dec 18 '18 at 1:51
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The 2-norm of a matrix is its maximal singular value? How to represent the singular value of A?
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– Kristy
Dec 18 '18 at 1:54
add a comment |
$begingroup$
$A$ is an $m×n$ matrix, verify the inequality $|A|_inftyleqsqrt n|A|_2$ and give an example of a nonzero matrix if the equality is achieved.
linear-algebra
asked Dec 18 '18 at 0:41
KristyKristy
454
$endgroup$
$A$ is an $m×n$ matrix, verify the inequality $|A|_inftyleqsqrt n|A|_2$ and give an example of a nonzero matrix if the equality is achieved.
linear-algebra
linear-algebra
asked Dec 18 '18 at 0:41
KristyKristy
454
asked Dec 18 '18 at 0:41
KristyKristy
454
asked Dec 18 '18 at 0:41
KristyKristy
454
asked Dec 18 '18 at 0:41
KristyKristy
454
asked Dec 18 '18 at 0:41
KristyKristy
454
454
closed as off-topic by Saad, Leucippus, Eevee Trainer, A Blumenthal, mrtaurho Dec 18 '18 at 5:39
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Leucippus, Eevee Trainer, A Blumenthal, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Saad, Leucippus, Eevee Trainer, A Blumenthal, mrtaurho Dec 18 '18 at 5:39
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Leucippus, Eevee Trainer, A Blumenthal, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
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Where are you stuck? Maybe start with reviewing the definitions of these norms.
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– Dave
Dec 18 '18 at 1:51
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The 2-norm of a matrix is its maximal singular value? How to represent the singular value of A?
$endgroup$
– Kristy
Dec 18 '18 at 1:54
add a comment |
$begingroup$
Where are you stuck? Maybe start with reviewing the definitions of these norms.
$endgroup$
– Dave
Dec 18 '18 at 1:51
$begingroup$
The 2-norm of a matrix is its maximal singular value? How to represent the singular value of A?
$endgroup$
– Kristy
Dec 18 '18 at 1:54
$begingroup$
Where are you stuck? Maybe start with reviewing the definitions of these norms.
$endgroup$
– Dave
Dec 18 '18 at 1:51
$begingroup$
Where are you stuck? Maybe start with reviewing the definitions of these norms.
$endgroup$
– Dave
Dec 18 '18 at 1:51
$begingroup$
The 2-norm of a matrix is its maximal singular value? How to represent the singular value of A?
$endgroup$
– Kristy
Dec 18 '18 at 1:54
$begingroup$
The 2-norm of a matrix is its maximal singular value? How to represent the singular value of A?
$endgroup$
– Kristy
Dec 18 '18 at 1:54
add a comment |
1 Answer
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For any $0neq xinmathbb R^n$, we have $$Vert xVert_2=sqrt{x_1^2+cdots+x_n^2}leq sqrt{nmax_{1leq jleq n}x_j^2}=sqrt{n}Vert xVert_infty$$ and $$Vert AxVert_infty=max_{1leq jleq n}|(Ax)_j|=sqrt{max_{1leq jleq n}(Ax)_j^2}leqsqrt{(Ax)_1^2+cdots+(Ax)_n^2}=Vert AxVert_2$$ which gives $$frac{Vert AxVert_infty}{Vert xVert_infty}leqsqrt{n}frac{Vert AxVert_2}{Vert xVert_2}$$
answered Dec 18 '18 at 2:10
DaveDave
9,05311033
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
For any $0neq xinmathbb R^n$, we have $$Vert xVert_2=sqrt{x_1^2+cdots+x_n^2}leq sqrt{nmax_{1leq jleq n}x_j^2}=sqrt{n}Vert xVert_infty$$ and $$Vert AxVert_infty=max_{1leq jleq n}|(Ax)_j|=sqrt{max_{1leq jleq n}(Ax)_j^2}leqsqrt{(Ax)_1^2+cdots+(Ax)_n^2}=Vert AxVert_2$$ which gives $$frac{Vert AxVert_infty}{Vert xVert_infty}leqsqrt{n}frac{Vert AxVert_2}{Vert xVert_2}$$
answered Dec 18 '18 at 2:10
DaveDave
9,05311033
$endgroup$
add a comment |
$begingroup$
For any $0neq xinmathbb R^n$, we have $$Vert xVert_2=sqrt{x_1^2+cdots+x_n^2}leq sqrt{nmax_{1leq jleq n}x_j^2}=sqrt{n}Vert xVert_infty$$ and $$Vert AxVert_infty=max_{1leq jleq n}|(Ax)_j|=sqrt{max_{1leq jleq n}(Ax)_j^2}leqsqrt{(Ax)_1^2+cdots+(Ax)_n^2}=Vert AxVert_2$$ which gives $$frac{Vert AxVert_infty}{Vert xVert_infty}leqsqrt{n}frac{Vert AxVert_2}{Vert xVert_2}$$
answered Dec 18 '18 at 2:10
DaveDave
9,05311033
$endgroup$
add a comment |
$begingroup$
For any $0neq xinmathbb R^n$, we have $$Vert xVert_2=sqrt{x_1^2+cdots+x_n^2}leq sqrt{nmax_{1leq jleq n}x_j^2}=sqrt{n}Vert xVert_infty$$ and $$Vert AxVert_infty=max_{1leq jleq n}|(Ax)_j|=sqrt{max_{1leq jleq n}(Ax)_j^2}leqsqrt{(Ax)_1^2+cdots+(Ax)_n^2}=Vert AxVert_2$$ which gives $$frac{Vert AxVert_infty}{Vert xVert_infty}leqsqrt{n}frac{Vert AxVert_2}{Vert xVert_2}$$
answered Dec 18 '18 at 2:10
DaveDave
9,05311033
$endgroup$
For any $0neq xinmathbb R^n$, we have $$Vert xVert_2=sqrt{x_1^2+cdots+x_n^2}leq sqrt{nmax_{1leq jleq n}x_j^2}=sqrt{n}Vert xVert_infty$$ and $$Vert AxVert_infty=max_{1leq jleq n}|(Ax)_j|=sqrt{max_{1leq jleq n}(Ax)_j^2}leqsqrt{(Ax)_1^2+cdots+(Ax)_n^2}=Vert AxVert_2$$ which gives $$frac{Vert AxVert_infty}{Vert xVert_infty}leqsqrt{n}frac{Vert AxVert_2}{Vert xVert_2}$$
answered Dec 18 '18 at 2:10
DaveDave
9,05311033
answered Dec 18 '18 at 2:10
DaveDave
9,05311033
answered Dec 18 '18 at 2:10
DaveDave
9,05311033
answered Dec 18 '18 at 2:10
DaveDave
9,05311033
9,05311033
add a comment |
add a comment |
$begingroup$
Where are you stuck? Maybe start with reviewing the definitions of these norms.
$endgroup$
– Dave
Dec 18 '18 at 1:51
$begingroup$
The 2-norm of a matrix is its maximal singular value? How to represent the singular value of A?
$endgroup$
– Kristy
Dec 18 '18 at 1:54