Relation between Frobenius, spectral norm and sum of maximaKind of submultiplicativity of the Frobenius norm:...
Relation between Frobenius, spectral norm and sum of maxima
Kind of submultiplicativity of the Frobenius norm: $|AB|_F leq |A|_2|B|_F$?Bounding sum of first singular values squared for Kronecker sum of traceless matricesOperator norm vs spectral radius for positive matricesBounds on the effect of a matrix product on the Frobenius normBounds on smallest Eigenvalue of the Sum of a Standard Laplacian and a Diagonal MatrixHow to calculate expected value of matrix norms of $A^TA$?Norm and trace inequalitiesMinimize spectral norm under diagonal similarityOperator norm of a soft thresholded symmetric matrixRelation between Frobenius norm, infinity norm and sum of maxima
3
$begingroup$
Let $A$ be a $n times n$ matrix so that the Frobenius norm squared $|A|_F^2$ is $Theta(n)$, the spectral norm squared $|A|_2^2=1$. Is it true that $sum_{i=1}^nmax_{1leq jleq n} |A_{ij}|^2$ is $Omega(n)$? Assume that $n$ is sufficiently large.
I cannot find a relation between matrix norms that can show this. The idea behind this question is that there are many singular values of $A$ that are $Theta(1)$.
Thanks!
linear-algebra matrices norms
edited 14 hours ago
Liviu Nicolaescu
25.9k260111
asked 19 hours ago
horxiohorxio
263
New contributor
horxio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
|
show 4 more comments
3
$begingroup$
Let $A$ be a $n times n$ matrix so that the Frobenius norm squared $|A|_F^2$ is $Theta(n)$, the spectral norm squared $|A|_2^2=1$. Is it true that $sum_{i=1}^nmax_{1leq jleq n} |A_{ij}|^2$ is $Omega(n)$? Assume that $n$ is sufficiently large.
I cannot find a relation between matrix norms that can show this. The idea behind this question is that there are many singular values of $A$ that are $Theta(1)$.
Thanks!
linear-algebra matrices norms
edited 14 hours ago
Liviu Nicolaescu
25.9k260111
asked 19 hours ago
horxiohorxio
263
New contributor
horxio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
2
$begingroup$
According to Wikipedia, $|A|_F=|A|_2$. Please, explain your notation!
$endgroup$
– W-t-P
19 hours ago
1
$begingroup$
Frobenius norm, where did you find that? It is wrong what you are saying.
$endgroup$
– horxio
19 hours ago
1
$begingroup$
What you are saying is incorrect. Can you please tell me where exactly you found this relation? It is wrong that the Frobenius norm is equal to the spectral norm. Think about it, if they were equal, why should we have two definitions? it holds that $||A||_F geq ||A||_2$.
$endgroup$
– horxio
18 hours ago
2
$begingroup$
The last sentence in the ""Entrywise" matrix norms" reads: The special case p = 2 is the Frobenius norm; see also the "Frobenius norm" section below. Instead of arguing who is (in)correct, please explain your notation.
$endgroup$
– W-t-P
18 hours ago
1
$begingroup$
@W-t-P I find your comments towards a new user a bit aggressive. The problem here is that $|A|_2$ is standard notation for two different things, as the Wikipedia page that you linked also notes (if you read a bit earlier, These norms again share the notation with the induced and entrywise p-norms, but they are different, and earlier the definition of the spectral norm). On the other hand, the terms Frobenius norm and spectral norm are unambiguous and look perfectly fine to me as explanations of the notation in OP's question.
$endgroup$
– Federico Poloni
14 hours ago
|
show 4 more comments
3
3
3
$begingroup$
Let $A$ be a $n times n$ matrix so that the Frobenius norm squared $|A|_F^2$ is $Theta(n)$, the spectral norm squared $|A|_2^2=1$. Is it true that $sum_{i=1}^nmax_{1leq jleq n} |A_{ij}|^2$ is $Omega(n)$? Assume that $n$ is sufficiently large.
I cannot find a relation between matrix norms that can show this. The idea behind this question is that there are many singular values of $A$ that are $Theta(1)$.
Thanks!
linear-algebra matrices norms
edited 14 hours ago
Liviu Nicolaescu
25.9k260111
asked 19 hours ago
horxiohorxio
263
New contributor
horxio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Let $A$ be a $n times n$ matrix so that the Frobenius norm squared $|A|_F^2$ is $Theta(n)$, the spectral norm squared $|A|_2^2=1$. Is it true that $sum_{i=1}^nmax_{1leq jleq n} |A_{ij}|^2$ is $Omega(n)$? Assume that $n$ is sufficiently large.
I cannot find a relation between matrix norms that can show this. The idea behind this question is that there are many singular values of $A$ that are $Theta(1)$.
Thanks!
linear-algebra matrices norms
linear-algebra matrices norms
edited 14 hours ago
Liviu Nicolaescu
25.9k260111
asked 19 hours ago
horxiohorxio
263
New contributor
horxio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 14 hours ago
Liviu Nicolaescu
25.9k260111
asked 19 hours ago
horxiohorxio
263
New contributor
horxio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 14 hours ago
Liviu Nicolaescu
25.9k260111
edited 14 hours ago
Liviu Nicolaescu
25.9k260111
edited 14 hours ago
Liviu Nicolaescu
25.9k260111
25.9k260111
asked 19 hours ago
horxiohorxio
263
New contributor
horxio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 19 hours ago
horxiohorxio
263
asked 19 hours ago
horxiohorxio
263
263
New contributor
horxio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
horxio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
horxio is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
$begingroup$
According to Wikipedia, $|A|_F=|A|_2$. Please, explain your notation!
$endgroup$
– W-t-P
19 hours ago
1
$begingroup$
Frobenius norm, where did you find that? It is wrong what you are saying.
$endgroup$
– horxio
19 hours ago
1
$begingroup$
What you are saying is incorrect. Can you please tell me where exactly you found this relation? It is wrong that the Frobenius norm is equal to the spectral norm. Think about it, if they were equal, why should we have two definitions? it holds that $||A||_F geq ||A||_2$.
$endgroup$
– horxio
18 hours ago
2
$begingroup$
The last sentence in the ""Entrywise" matrix norms" reads: The special case p = 2 is the Frobenius norm; see also the "Frobenius norm" section below. Instead of arguing who is (in)correct, please explain your notation.
$endgroup$
– W-t-P
18 hours ago
1
$begingroup$
@W-t-P I find your comments towards a new user a bit aggressive. The problem here is that $|A|_2$ is standard notation for two different things, as the Wikipedia page that you linked also notes (if you read a bit earlier, These norms again share the notation with the induced and entrywise p-norms, but they are different, and earlier the definition of the spectral norm). On the other hand, the terms Frobenius norm and spectral norm are unambiguous and look perfectly fine to me as explanations of the notation in OP's question.
$endgroup$
– Federico Poloni
14 hours ago
|
show 4 more comments
2
$begingroup$
According to Wikipedia, $|A|_F=|A|_2$. Please, explain your notation!
$endgroup$
– W-t-P
19 hours ago
1
$begingroup$
Frobenius norm, where did you find that? It is wrong what you are saying.
$endgroup$
– horxio
19 hours ago
1
$begingroup$
What you are saying is incorrect. Can you please tell me where exactly you found this relation? It is wrong that the Frobenius norm is equal to the spectral norm. Think about it, if they were equal, why should we have two definitions? it holds that $||A||_F geq ||A||_2$.
$endgroup$
– horxio
18 hours ago
2
$begingroup$
The last sentence in the ""Entrywise" matrix norms" reads: The special case p = 2 is the Frobenius norm; see also the "Frobenius norm" section below. Instead of arguing who is (in)correct, please explain your notation.
$endgroup$
– W-t-P
18 hours ago
1
$begingroup$
@W-t-P I find your comments towards a new user a bit aggressive. The problem here is that $|A|_2$ is standard notation for two different things, as the Wikipedia page that you linked also notes (if you read a bit earlier, These norms again share the notation with the induced and entrywise p-norms, but they are different, and earlier the definition of the spectral norm). On the other hand, the terms Frobenius norm and spectral norm are unambiguous and look perfectly fine to me as explanations of the notation in OP's question.
$endgroup$
– Federico Poloni
14 hours ago
2
2
$begingroup$
According to Wikipedia, $|A|_F=|A|_2$. Please, explain your notation!
$endgroup$
– W-t-P
19 hours ago
$begingroup$
According to Wikipedia, $|A|_F=|A|_2$. Please, explain your notation!
$endgroup$
– W-t-P
19 hours ago
1
1
$begingroup$
Frobenius norm, where did you find that? It is wrong what you are saying.
$endgroup$
– horxio
19 hours ago
$begingroup$
Frobenius norm, where did you find that? It is wrong what you are saying.
$endgroup$
– horxio
19 hours ago
1
1
$begingroup$
What you are saying is incorrect. Can you please tell me where exactly you found this relation? It is wrong that the Frobenius norm is equal to the spectral norm. Think about it, if they were equal, why should we have two definitions? it holds that $||A||_F geq ||A||_2$.
$endgroup$
– horxio
18 hours ago
$begingroup$
What you are saying is incorrect. Can you please tell me where exactly you found this relation? It is wrong that the Frobenius norm is equal to the spectral norm. Think about it, if they were equal, why should we have two definitions? it holds that $||A||_F geq ||A||_2$.
$endgroup$
– horxio
18 hours ago
2
2
$begingroup$
The last sentence in the ""Entrywise" matrix norms" reads: The special case p = 2 is the Frobenius norm; see also the "Frobenius norm" section below. Instead of arguing who is (in)correct, please explain your notation.
$endgroup$
– W-t-P
18 hours ago
$begingroup$
The last sentence in the ""Entrywise" matrix norms" reads: The special case p = 2 is the Frobenius norm; see also the "Frobenius norm" section below. Instead of arguing who is (in)correct, please explain your notation.
$endgroup$
– W-t-P
18 hours ago
1
1
$begingroup$
@W-t-P I find your comments towards a new user a bit aggressive. The problem here is that $|A|_2$ is standard notation for two different things, as the Wikipedia page that you linked also notes (if you read a bit earlier, These norms again share the notation with the induced and entrywise p-norms, but they are different, and earlier the definition of the spectral norm). On the other hand, the terms Frobenius norm and spectral norm are unambiguous and look perfectly fine to me as explanations of the notation in OP's question.
$endgroup$
– Federico Poloni
14 hours ago
$begingroup$
@W-t-P I find your comments towards a new user a bit aggressive. The problem here is that $|A|_2$ is standard notation for two different things, as the Wikipedia page that you linked also notes (if you read a bit earlier, These norms again share the notation with the induced and entrywise p-norms, but they are different, and earlier the definition of the spectral norm). On the other hand, the terms Frobenius norm and spectral norm are unambiguous and look perfectly fine to me as explanations of the notation in OP's question.
$endgroup$
– Federico Poloni
14 hours ago
|
show 4 more comments
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2
$begingroup$
According to Wikipedia, $|A|_F=|A|_2$. Please, explain your notation!
$endgroup$
– W-t-P
19 hours ago
1
$begingroup$
Frobenius norm, where did you find that? It is wrong what you are saying.
$endgroup$
– horxio
19 hours ago
1
$begingroup$
What you are saying is incorrect. Can you please tell me where exactly you found this relation? It is wrong that the Frobenius norm is equal to the spectral norm. Think about it, if they were equal, why should we have two definitions? it holds that $||A||_F geq ||A||_2$.
$endgroup$
– horxio
18 hours ago
2
$begingroup$
The last sentence in the ""Entrywise" matrix norms" reads: The special case p = 2 is the Frobenius norm; see also the "Frobenius norm" section below. Instead of arguing who is (in)correct, please explain your notation.
$endgroup$
– W-t-P
18 hours ago
1
$begingroup$
@W-t-P I find your comments towards a new user a bit aggressive. The problem here is that $|A|_2$ is standard notation for two different things, as the Wikipedia page that you linked also notes (if you read a bit earlier, These norms again share the notation with the induced and entrywise p-norms, but they are different, and earlier the definition of the spectral norm). On the other hand, the terms Frobenius norm and spectral norm are unambiguous and look perfectly fine to me as explanations of the notation in OP's question.
$endgroup$
– Federico Poloni
14 hours ago