Can the Induced Matrix Norm be Proven with Triangle Inequality?

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Discussion Overview

The discussion revolves around the proof of the statement $$||A^k|| \leq||A||^{k}$$ for a matrix A and a positive integer k, specifically focusing on the properties of induced matrix norms and the application of the triangle inequality. Participants explore the implications of the definitions of matrix norms and the supremum involved in the context of this inequality.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant expresses uncertainty about the proof of the inequality involving induced matrix norms and suggests that the triangle inequality might be relevant.
  • Another participant questions whether the maximizer x for $$||A^k||$$ is the same as that for $$||A||$$, indicating a potential issue with the proof's assumptions.
  • A further clarification is provided regarding the use of a specific vector x and its implications for the inequality, suggesting that the relationship holds through induction.
  • Some participants discuss the definition of norms and the supremum, with one noting that the definition of $$||AX||$$ as $$||A||$$ may not be universally applicable without assuming a maximum exists.
  • There is a recognition of the complexity of the concepts involved, with one participant expressing a desire to improve their understanding of similar problems in the future.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of certain statements regarding the norms and their implications. There are multiple competing views on the assumptions necessary for the proof and the definitions of the norms involved.

Contextual Notes

Participants highlight the need for careful consideration of definitions and assumptions, particularly regarding the supremum and the conditions under which the inequalities hold. There is an acknowledgment that the proof may depend on the existence of a maximum for the norms discussed.

FOIWATER
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Hi, I found a statement without a proof. It seems simple enough, but I am having trouble proving it because I am not positive about induced matrix norms. The statement is that $$||A^k|| \leq||A||^{k}$$ for some matrix A and positive integer k. I have found that the norm of a matrix is the supremum of the norm of Ax over the norm of x, but I do not know to which norms these refer?

I am assuming euclidean norms. Since Ax gives us back a vector and x is itself a vector.

So I have that:
$$||A^k|| = \sup_{||x||=1}(||A^{k}x|| : ||x||=1)$$
and
$$||A||^{k} = \sup_{||x||=1}(||Ax|| : ||x||=1)^{k}$$

Not sure what to do with them, though, any hints appreciated. I was thinking triangle inequality.. but I didn't really get anything from it. And I do not know if the triangle inequality applies to this matrix induced norm (although I think it applies to any operation that qualifies as a norm, since it defines norms).
 
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What if the x that gives the sup for A^k is not the same as the x that is the sup for A? A acting on any vector will always be less than or equal to its sup.
 
To be more clear...
Let ##X## be your maximizer for ##\|Ax\|##. Then
##\| A^k \| = \| A A A ... A \| = \sup_{\|x\|=1} \left( \| A A A ... Ax \| \right)##
Let ##x_0 ## be any x such that ##\|x_0 \| = 1##.
Then you know that ##\|Ax_0\| \leq \|AX\|=\|A\|. ##
Let ##x_1 = Ax_0##. By similar argument, ##\|A\frac{x_1}{\|x_1\|}\|=\frac{\|Ax_1\|}{\|x_1\|} \leq \|AX\|\implies \|Ax_1\|\leq \|AX\|\|x_1\| \leq \|AX\|\|A\|=\|A\|^2 .##
And by induction, you can clearly see that no matter what, you won't be able to break the inequality.
 
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∥AX∥=∥A∥ I am not sure of this statement? A times X should give a vector, not a matrix A?
 
This actually makes sense now thanks a lot!
 
Last edited:
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FOIWATER said:
∥AX∥=∥A∥ I am not sure of this statement? A times X should give a vector, not a matrix A?
Right, but I defined ||AX|| to be ||A|| earlier.
RUber said:
Let X be your maximizer for ∥Ax∥
Of course, that is assuming that there is a maximum. You might have to keep the definition of ||A|| as the supremum to be proper...but the logic is the same.
 
That's good insight I hope to be able to work my way through problems like that some day
 

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