Proof of infinity matrix norm
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Given the $l_{infty}$ matrix norm for $A{in}{Bbb{R}}^{mxn}$ is defined as: $|A|_{infty} =max_{1 leq i leq n}|a^{i}|_{1}$ (where $a^{i}$ is the i$^{th}$) row in matrix A),
Show that:
$|A|_{infty} =max left{|Ax|_{infty} : x_{infty} le 1right} =max left{|Ax|_{infty} : x_{infty} = 1right}$
I know that this is a property of subordinate matrix norms but I'm not sure how to go about with proving it.
lp-spaces matrix-norms
asked Jan 24 at 16:38
AndyAndy
1
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add a comment |
$begingroup$
Given the $l_{infty}$ matrix norm for $A{in}{Bbb{R}}^{mxn}$ is defined as: $|A|_{infty} =max_{1 leq i leq n}|a^{i}|_{1}$ (where $a^{i}$ is the i$^{th}$) row in matrix A),
Show that:
$|A|_{infty} =max left{|Ax|_{infty} : x_{infty} le 1right} =max left{|Ax|_{infty} : x_{infty} = 1right}$
I know that this is a property of subordinate matrix norms but I'm not sure how to go about with proving it.
lp-spaces matrix-norms
asked Jan 24 at 16:38
AndyAndy
1
$endgroup$
$begingroup$
Welcome to Math.SE. To begin you might consider inequalities among the three expressions which you are able to prove, e.g. $max left{|Ax|_{infty} : x_{infty} le 1right} ge max left{|Ax|_{infty} : x_{infty} = 1right}$.
$endgroup$
– hardmath
Jan 24 at 16:44
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Does $||a^i||_1 = sum_{j=1}^n |a^i_j|$? i.e. the sum of absolute values of the elements of the $a^i$?
$endgroup$
– астон вілла олоф мэллбэрг
Jan 24 at 16:44
$begingroup$
Yes, it's the $l_{1}$ norm of the vector $a^{i}$. Should have clarified that.
$endgroup$
– Andy
Jan 24 at 16:46
add a comment |
$begingroup$
Given the $l_{infty}$ matrix norm for $A{in}{Bbb{R}}^{mxn}$ is defined as: $|A|_{infty} =max_{1 leq i leq n}|a^{i}|_{1}$ (where $a^{i}$ is the i$^{th}$) row in matrix A),
Show that:
$|A|_{infty} =max left{|Ax|_{infty} : x_{infty} le 1right} =max left{|Ax|_{infty} : x_{infty} = 1right}$
I know that this is a property of subordinate matrix norms but I'm not sure how to go about with proving it.
lp-spaces matrix-norms
asked Jan 24 at 16:38
AndyAndy
1
$endgroup$
Given the $l_{infty}$ matrix norm for $A{in}{Bbb{R}}^{mxn}$ is defined as: $|A|_{infty} =max_{1 leq i leq n}|a^{i}|_{1}$ (where $a^{i}$ is the i$^{th}$) row in matrix A),
Show that:
$|A|_{infty} =max left{|Ax|_{infty} : x_{infty} le 1right} =max left{|Ax|_{infty} : x_{infty} = 1right}$
I know that this is a property of subordinate matrix norms but I'm not sure how to go about with proving it.
lp-spaces matrix-norms
lp-spaces matrix-norms
asked Jan 24 at 16:38
AndyAndy
1
asked Jan 24 at 16:38
AndyAndy
1
asked Jan 24 at 16:38
AndyAndy
1
asked Jan 24 at 16:38
AndyAndy
1
asked Jan 24 at 16:38
AndyAndy
1
1
$begingroup$
Welcome to Math.SE. To begin you might consider inequalities among the three expressions which you are able to prove, e.g. $max left{|Ax|_{infty} : x_{infty} le 1right} ge max left{|Ax|_{infty} : x_{infty} = 1right}$.
$endgroup$
– hardmath
Jan 24 at 16:44
$begingroup$
Does $||a^i||_1 = sum_{j=1}^n |a^i_j|$? i.e. the sum of absolute values of the elements of the $a^i$?
$endgroup$
– астон вілла олоф мэллбэрг
Jan 24 at 16:44
$begingroup$
Yes, it's the $l_{1}$ norm of the vector $a^{i}$. Should have clarified that.
$endgroup$
– Andy
Jan 24 at 16:46
add a comment |
$begingroup$
Welcome to Math.SE. To begin you might consider inequalities among the three expressions which you are able to prove, e.g. $max left{|Ax|_{infty} : x_{infty} le 1right} ge max left{|Ax|_{infty} : x_{infty} = 1right}$.
$endgroup$
– hardmath
Jan 24 at 16:44
$begingroup$
Does $||a^i||_1 = sum_{j=1}^n |a^i_j|$? i.e. the sum of absolute values of the elements of the $a^i$?
$endgroup$
– астон вілла олоф мэллбэрг
Jan 24 at 16:44
$begingroup$
Yes, it's the $l_{1}$ norm of the vector $a^{i}$. Should have clarified that.
$endgroup$
– Andy
Jan 24 at 16:46
$begingroup$
Welcome to Math.SE. To begin you might consider inequalities among the three expressions which you are able to prove, e.g. $max left{|Ax|_{infty} : x_{infty} le 1right} ge max left{|Ax|_{infty} : x_{infty} = 1right}$.
$endgroup$
– hardmath
Jan 24 at 16:44
$begingroup$
Welcome to Math.SE. To begin you might consider inequalities among the three expressions which you are able to prove, e.g. $max left{|Ax|_{infty} : x_{infty} le 1right} ge max left{|Ax|_{infty} : x_{infty} = 1right}$.
$endgroup$
– hardmath
Jan 24 at 16:44
$begingroup$
Does $||a^i||_1 = sum_{j=1}^n |a^i_j|$? i.e. the sum of absolute values of the elements of the $a^i$?
$endgroup$
– астон вілла олоф мэллбэрг
Jan 24 at 16:44
$begingroup$
Does $||a^i||_1 = sum_{j=1}^n |a^i_j|$? i.e. the sum of absolute values of the elements of the $a^i$?
$endgroup$
– астон вілла олоф мэллбэрг
Jan 24 at 16:44
$begingroup$
Yes, it's the $l_{1}$ norm of the vector $a^{i}$. Should have clarified that.
$endgroup$
– Andy
Jan 24 at 16:46
$begingroup$
Yes, it's the $l_{1}$ norm of the vector $a^{i}$. Should have clarified that.
$endgroup$
– Andy
Jan 24 at 16:46
add a comment |
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$begingroup$
Welcome to Math.SE. To begin you might consider inequalities among the three expressions which you are able to prove, e.g. $max left{|Ax|_{infty} : x_{infty} le 1right} ge max left{|Ax|_{infty} : x_{infty} = 1right}$.
$endgroup$
– hardmath
Jan 24 at 16:44
$begingroup$
Does $||a^i||_1 = sum_{j=1}^n |a^i_j|$? i.e. the sum of absolute values of the elements of the $a^i$?
$endgroup$
– астон вілла олоф мэллбэрг
Jan 24 at 16:44
$begingroup$
Yes, it's the $l_{1}$ norm of the vector $a^{i}$. Should have clarified that.
$endgroup$
– Andy
Jan 24 at 16:46