Is there a matrix norm induced by vector norm for 0<p<1?
up vote
1
down vote
favorite
A matrix norm induced by vector norm defined here is $||A||_p$ with $pgeq 1$.
But how is matrix norm defined for $p<1$ ?
Edit:
I am asking this because in the second half of below theorem(in the image), where $r_sigma (A)$ represents spectral radius of A, $epsilon >0$ may be between 0 and 1. How is matrix norm defined then ?
linear-algebra
asked Nov 18 at 11:34
Ganesh Gani
307
add a comment |
up vote
1
down vote
favorite
A matrix norm induced by vector norm defined here is $||A||_p$ with $pgeq 1$.
But how is matrix norm defined for $p<1$ ?
Edit:
I am asking this because in the second half of below theorem(in the image), where $r_sigma (A)$ represents spectral radius of A, $epsilon >0$ may be between 0 and 1. How is matrix norm defined then ?
linear-algebra
asked Nov 18 at 11:34
Ganesh Gani
307
1
The author probably doesn't mean that $|cdot|_epsilon$ is induced from $left(sum_{k=1}^n |x_k|^epsilonright)^{1/epsilon}$. What he/she means are most likely that (a) $|cdot|_epsilon$ is induced from some vector norm, and (b) for the given $A$, the inequality (7.3.24) holds. The $epsilon$ in $|cdot|_epsilon$ here is just a subscript. It doesn't indicate what vector norm the matrix norm is induced from. I don't know the context of the quoted passage, but it looks like a prelude to Gelfand's formula.
– user1551
Nov 18 at 14:27
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
A matrix norm induced by vector norm defined here is $||A||_p$ with $pgeq 1$.
But how is matrix norm defined for $p<1$ ?
Edit:
I am asking this because in the second half of below theorem(in the image), where $r_sigma (A)$ represents spectral radius of A, $epsilon >0$ may be between 0 and 1. How is matrix norm defined then ?
linear-algebra
asked Nov 18 at 11:34
Ganesh Gani
307
A matrix norm induced by vector norm defined here is $||A||_p$ with $pgeq 1$.
But how is matrix norm defined for $p<1$ ?
Edit:
I am asking this because in the second half of below theorem(in the image), where $r_sigma (A)$ represents spectral radius of A, $epsilon >0$ may be between 0 and 1. How is matrix norm defined then ?
linear-algebra
linear-algebra
asked Nov 18 at 11:34
Ganesh Gani
307
asked Nov 18 at 11:34
Ganesh Gani
307
edited Nov 18 at 12:30
asked Nov 18 at 11:34
Ganesh Gani
307
asked Nov 18 at 11:34
Ganesh Gani
307
asked Nov 18 at 11:34
Ganesh Gani
307
307
1
The author probably doesn't mean that $|cdot|_epsilon$ is induced from $left(sum_{k=1}^n |x_k|^epsilonright)^{1/epsilon}$. What he/she means are most likely that (a) $|cdot|_epsilon$ is induced from some vector norm, and (b) for the given $A$, the inequality (7.3.24) holds. The $epsilon$ in $|cdot|_epsilon$ here is just a subscript. It doesn't indicate what vector norm the matrix norm is induced from. I don't know the context of the quoted passage, but it looks like a prelude to Gelfand's formula.
– user1551
Nov 18 at 14:27
add a comment |
1
The author probably doesn't mean that $|cdot|_epsilon$ is induced from $left(sum_{k=1}^n |x_k|^epsilonright)^{1/epsilon}$. What he/she means are most likely that (a) $|cdot|_epsilon$ is induced from some vector norm, and (b) for the given $A$, the inequality (7.3.24) holds. The $epsilon$ in $|cdot|_epsilon$ here is just a subscript. It doesn't indicate what vector norm the matrix norm is induced from. I don't know the context of the quoted passage, but it looks like a prelude to Gelfand's formula.
– user1551
Nov 18 at 14:27
1
1
The author probably doesn't mean that $|cdot|_epsilon$ is induced from $left(sum_{k=1}^n |x_k|^epsilonright)^{1/epsilon}$. What he/she means are most likely that (a) $|cdot|_epsilon$ is induced from some vector norm, and (b) for the given $A$, the inequality (7.3.24) holds. The $epsilon$ in $|cdot|_epsilon$ here is just a subscript. It doesn't indicate what vector norm the matrix norm is induced from. I don't know the context of the quoted passage, but it looks like a prelude to Gelfand's formula.
– user1551
Nov 18 at 14:27
The author probably doesn't mean that $|cdot|_epsilon$ is induced from $left(sum_{k=1}^n |x_k|^epsilonright)^{1/epsilon}$. What he/she means are most likely that (a) $|cdot|_epsilon$ is induced from some vector norm, and (b) for the given $A$, the inequality (7.3.24) holds. The $epsilon$ in $|cdot|_epsilon$ here is just a subscript. It doesn't indicate what vector norm the matrix norm is induced from. I don't know the context of the quoted passage, but it looks like a prelude to Gelfand's formula.
– user1551
Nov 18 at 14:27
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
For vectors $x in mathbb{K}^n$ (with $mathbb{K}in {mathbb{R},mathbb{C}}$) the $p$-Norm $lVert cdot rVert_p$ is as well only defined for $p geq 1$.
The reason is, that for $p<1$ the usual definition $$lVert x rVert_p := left(sum_{k=1}^n |x_k|^pright)^{1/p}$$ doesn't result in a norm.
answered Nov 18 at 11:46
bruderjakob17
1286
I have added few more things in question.
– Ganesh Gani
Nov 18 at 12:34
What does matrix norm $||A||_epsilon$ represent when $0<epsilon <1$ ?
– Ganesh Gani
Nov 18 at 12:36
Is there given a proof to the theorem?
– bruderjakob17
Nov 18 at 12:40
No. Not given in that material. Its proof is a nontrivial construction, and a proof of it is given in Isaacson and Keller (1966, p. 12)
– Ganesh Gani
Nov 18 at 13:47
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
For vectors $x in mathbb{K}^n$ (with $mathbb{K}in {mathbb{R},mathbb{C}}$) the $p$-Norm $lVert cdot rVert_p$ is as well only defined for $p geq 1$.
The reason is, that for $p<1$ the usual definition $$lVert x rVert_p := left(sum_{k=1}^n |x_k|^pright)^{1/p}$$ doesn't result in a norm.
answered Nov 18 at 11:46
bruderjakob17
1286
I have added few more things in question.
– Ganesh Gani
Nov 18 at 12:34
What does matrix norm $||A||_epsilon$ represent when $0<epsilon <1$ ?
– Ganesh Gani
Nov 18 at 12:36
Is there given a proof to the theorem?
– bruderjakob17
Nov 18 at 12:40
No. Not given in that material. Its proof is a nontrivial construction, and a proof of it is given in Isaacson and Keller (1966, p. 12)
– Ganesh Gani
Nov 18 at 13:47
add a comment |
up vote
0
down vote
For vectors $x in mathbb{K}^n$ (with $mathbb{K}in {mathbb{R},mathbb{C}}$) the $p$-Norm $lVert cdot rVert_p$ is as well only defined for $p geq 1$.
The reason is, that for $p<1$ the usual definition $$lVert x rVert_p := left(sum_{k=1}^n |x_k|^pright)^{1/p}$$ doesn't result in a norm.
answered Nov 18 at 11:46
bruderjakob17
1286
I have added few more things in question.
– Ganesh Gani
Nov 18 at 12:34
What does matrix norm $||A||_epsilon$ represent when $0<epsilon <1$ ?
– Ganesh Gani
Nov 18 at 12:36
Is there given a proof to the theorem?
– bruderjakob17
Nov 18 at 12:40
No. Not given in that material. Its proof is a nontrivial construction, and a proof of it is given in Isaacson and Keller (1966, p. 12)
– Ganesh Gani
Nov 18 at 13:47
add a comment |
up vote
0
down vote
up vote
0
down vote
For vectors $x in mathbb{K}^n$ (with $mathbb{K}in {mathbb{R},mathbb{C}}$) the $p$-Norm $lVert cdot rVert_p$ is as well only defined for $p geq 1$.
The reason is, that for $p<1$ the usual definition $$lVert x rVert_p := left(sum_{k=1}^n |x_k|^pright)^{1/p}$$ doesn't result in a norm.
answered Nov 18 at 11:46
bruderjakob17
1286
For vectors $x in mathbb{K}^n$ (with $mathbb{K}in {mathbb{R},mathbb{C}}$) the $p$-Norm $lVert cdot rVert_p$ is as well only defined for $p geq 1$.
The reason is, that for $p<1$ the usual definition $$lVert x rVert_p := left(sum_{k=1}^n |x_k|^pright)^{1/p}$$ doesn't result in a norm.
answered Nov 18 at 11:46
bruderjakob17
1286
answered Nov 18 at 11:46
bruderjakob17
1286
answered Nov 18 at 11:46
bruderjakob17
1286
answered Nov 18 at 11:46
bruderjakob17
1286
1286
I have added few more things in question.
– Ganesh Gani
Nov 18 at 12:34
What does matrix norm $||A||_epsilon$ represent when $0<epsilon <1$ ?
– Ganesh Gani
Nov 18 at 12:36
Is there given a proof to the theorem?
– bruderjakob17
Nov 18 at 12:40
No. Not given in that material. Its proof is a nontrivial construction, and a proof of it is given in Isaacson and Keller (1966, p. 12)
– Ganesh Gani
Nov 18 at 13:47
add a comment |
I have added few more things in question.
– Ganesh Gani
Nov 18 at 12:34
What does matrix norm $||A||_epsilon$ represent when $0<epsilon <1$ ?
– Ganesh Gani
Nov 18 at 12:36
Is there given a proof to the theorem?
– bruderjakob17
Nov 18 at 12:40
No. Not given in that material. Its proof is a nontrivial construction, and a proof of it is given in Isaacson and Keller (1966, p. 12)
– Ganesh Gani
Nov 18 at 13:47
I have added few more things in question.
– Ganesh Gani
Nov 18 at 12:34
I have added few more things in question.
– Ganesh Gani
Nov 18 at 12:34
What does matrix norm $||A||_epsilon$ represent when $0<epsilon <1$ ?
– Ganesh Gani
Nov 18 at 12:36
What does matrix norm $||A||_epsilon$ represent when $0<epsilon <1$ ?
– Ganesh Gani
Nov 18 at 12:36
Is there given a proof to the theorem?
– bruderjakob17
Nov 18 at 12:40
Is there given a proof to the theorem?
– bruderjakob17
Nov 18 at 12:40
No. Not given in that material. Its proof is a nontrivial construction, and a proof of it is given in Isaacson and Keller (1966, p. 12)
– Ganesh Gani
Nov 18 at 13:47
No. Not given in that material. Its proof is a nontrivial construction, and a proof of it is given in Isaacson and Keller (1966, p. 12)
– Ganesh Gani
Nov 18 at 13:47
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003428%2fis-there-a-matrix-norm-induced-by-vector-norm-for-0p1%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
The author probably doesn't mean that $|cdot|_epsilon$ is induced from $left(sum_{k=1}^n |x_k|^epsilonright)^{1/epsilon}$. What he/she means are most likely that (a) $|cdot|_epsilon$ is induced from some vector norm, and (b) for the given $A$, the inequality (7.3.24) holds. The $epsilon$ in $|cdot|_epsilon$ here is just a subscript. It doesn't indicate what vector norm the matrix norm is induced from. I don't know the context of the quoted passage, but it looks like a prelude to Gelfand's formula.
– user1551
Nov 18 at 14:27