• Support PF! Buy your school textbooks, materials and every day products Here!

Infinity Norms and One Norms

  • Thread starter twoski
  • Start date
  • #1
181
2

Homework Statement



Determine constants c and C that do not depend on vector x but may depend on the dimension n of x, such that

[itex]c ||x||_{∞} ≤ ||x||_{1} ≤ C||x||_{∞}[/itex]

Use this result and the definition of matrix norms to find k and K that don't depend on the entries of matrix A (but depend on dimension n of A) such that

[itex]k ||A||_{∞} ≤ ||A||_{1} ≤ K||A||_{∞}[/itex]

The Attempt at a Solution



Firstly i'm still trying to wrap my head around what an infinity norm actually is. According to my lecture notes,

[itex] ||x||_{∞} \equiv 1 ≤ k ≤ n, max|x_{k}|[/itex]

Does this mean the infinity norm is just the largest absolute value in x? There is no summation sign so my best guess is that "max" refers to the highest possible value contained in x.

So in other words i need to figure out constants which satisfy the inequality. My first guess would be to take c=0 and C=1, assuming that infinity norms are always greater than one-norms.
 

Answers and Replies

  • #2
33,516
5,196

Homework Statement



Determine constants c and C that do not depend on vector x but may depend on the dimension n of x, such that

[itex]c ||x||_{∞} ≤ ||x||_{1} ≤ C||x||_{∞}[/itex]

Use this result and the definition of matrix norms to find k and K that don't depend on the entries of matrix A (but depend on dimension n of A) such that

[itex]k ||A||_{∞} ≤ ||A||_{1} ≤ K||A||_{∞}[/itex]

The Attempt at a Solution



Firstly i'm still trying to wrap my head around what an infinity norm actually is. According to my lecture notes,

[itex] ||x||_{∞} \equiv 1 ≤ k ≤ n, max|x_{k}|[/itex]

Does this mean the infinity norm is just the largest absolute value in x?
Sort of. It's the largest component xkof a vector x.
There is no summation sign so my best guess is that "max" refers to the highest possible value contained in x.

So in other words i need to figure out constants which satisfy the inequality. My first guess would be to take c=0 and C=1, assuming that infinity norms are always greater than one-norms.
 
  • #3
pasmith
Homework Helper
1,740
412

Homework Statement



Determine constants c and C that do not depend on vector x but may depend on the dimension n of x, such that

[itex]c ||x||_{∞} ≤ ||x||_{1} ≤ C||x||_{∞}[/itex]

Use this result and the definition of matrix norms to find k and K that don't depend on the entries of matrix A (but depend on dimension n of A) such that

[itex]k ||A||_{∞} ≤ ||A||_{1} ≤ K||A||_{∞}[/itex]

The Attempt at a Solution



Firstly i'm still trying to wrap my head around what an infinity norm actually is. According to my lecture notes,

[itex] ||x||_{∞} \equiv 1 ≤ k ≤ n, max|x_{k}|[/itex]

Does this mean the infinity norm is just the largest absolute value in x? There is no summation sign so my best guess is that "max" refers to the highest possible value contained in x.

So in other words i need to figure out constants which satisfy the inequality. My first guess would be to take c=0 and C=1, assuming that infinity norms are always greater than one-norms.
If [itex]x = (1,1)[/itex], what are [itex]\|x\|_1[/itex] and [itex]\|x\|_{\infty}[/itex]?

Taking [itex]c = 0[/itex] doesn't tell you anything; norms are non-negative by definition.

Hint for [itex]C[/itex]: given that [itex]\|x\|_1 = \sum_{i=1}^n {|x_i|}[/itex], what happens if you replace each summand with [itex]\|x\|_\infty = \max\{|x_j| : j = 1, \dots, n\}[/itex]?
 
  • #4
181
2
If [itex]x = (1,1)[/itex], what are [itex]\|x\|_1[/itex] and [itex]\|x\|_{\infty}[/itex]?

Taking [itex]c = 0[/itex] doesn't tell you anything; norms are non-negative by definition.

Hint for [itex]C[/itex]: given that [itex]\|x\|_1 = \sum_{i=1}^n {|x_i|}[/itex], what happens if you replace each summand with [itex]\|x\|_\infty = \max\{|x_j| : j = 1, \dots, n\}[/itex]?
[itex]\|x\|_1[/itex] would be 2 since you're just summing the absolute values contained in x (right?) and [itex]\|x\|_{\infty}[/itex] would be 1, i think.

I'm not sure i follow your second bit about C. If i were to replace every instance of |X|i with |X|∞ then, say for example, if i had x={1,2,3) I would be doing 3+3+3 instead of 1+2+3.
 
  • #5
pasmith
Homework Helper
1,740
412
[itex]\|x\|_1[/itex] would be 2 since you're just summing the absolute values contained in x (right?) and [itex]\|x\|_{\infty}[/itex] would be 1, i think.

I'm not sure i follow your second bit about C. If i were to replace every instance of |X|i with |X|∞ then, say for example, if i had x={1,2,3) I would be doing 3+3+3 instead of 1+2+3.
And is [itex]3 + 3 + 3 = 3\|(1,2,3)\|_{\infty}[/itex] greater than or less than [itex]1 + 2 + 3 = \|(1,2,3)\|_1[/itex]?
 
  • #6
181
2
The left hand side evaluates to 9 and the right hand side evaluates to 6. I find the question confusing because you could hypothetically find any 2 values of C and c which satisfy the question if you know what x contains. If x=(1,2,3) then you just need to pick C = 3, c = 1.

I assumed picking c=0 would make sense because the left hand side would always be zero if you're multiplying the infinity norm by zero. That would satisfy the left hand side, and the right hand side would just require a sufficiently large value for C to be greater than the one norm.
 
  • #7
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
3,750
99
Suppose that my vector x = (M,x2,...,xn) where M is larger than all the other entries (so is the infinity norm of x). Then:
[tex] ||x||_1 = |M| + |x_2| + ... + |x_n| \leq ???? [/tex]

The ????? should be writeable in terms of only M and the dimension of the space. Think about doing the same kind of procedure to this sum as you did with that (1,2,3) vector.

That is the harder direction. The easier one (I think) is to figure out
[tex] ???? \leq |M| + |x_2| + ... + |x_n|[/tex]
where again ??? is something that depends only on M and the dimension of the space
 
  • #8
181
2
Suppose that my vector x = (M,x2,...,xn) where M is larger than all the other entries (so is the infinity norm of x). Then:
[tex] ||x||_1 = |M| + |x_2| + ... + |x_n| \leq ???? [/tex]

The ????? should be writeable in terms of only M and the dimension of the space. Think about doing the same kind of procedure to this sum as you did with that (1,2,3) vector.

That is the harder direction. The easier one (I think) is to figure out
[tex] ???? \leq |M| + |x_2| + ... + |x_n|[/tex]
where again ??? is something that depends only on M and the dimension of the space
In terms of n (where n is the size of x), would this be correct for the first part?

[tex] ||x||_1 = |M| + |x_2| + ... + |x_n| \leq n * |M| [/tex]

For the second part, would i simply use

[tex] |M| * n-1 \leq |M| + |x_2| + ... + |x_n|[/tex]
 
  • #9
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
3,750
99
In terms of n (where n is the size of x), would this be correct for the first part?

[tex] ||x||_1 = |M| + |x_2| + ... + |x_n| \leq n * |M| [/tex]
This looks good (and should tell you what C is in the first part)

For the second part, would i simply use

[tex] |M| * n-1 \leq |M| + |x_2| + ... + |x_n|[/tex]
What if M = 1, x2 = x3 = 0 and n=3? You just told me that 3-1 = 2 < 1+0+0
 
  • #10
181
2
Hmmm, so if C=n then could c just be 1? There isn't much that the infinity norm could be multiplied by that would make it less than the one norm, right?
 
Last edited:

Related Threads on Infinity Norms and One Norms

  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
7
Views
7K
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
0
Views
4K
  • Last Post
Replies
0
Views
4K
Replies
0
Views
2K
  • Last Post
Replies
5
Views
690
  • Last Post
Replies
1
Views
312
  • Last Post
Replies
2
Views
759
  • Last Post
Replies
4
Views
1K
Top