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Linear Algebra

Let n∈Nn\in\mathbb{N}. Show that there exists a constant C (depending only n) such that if A∈Rnxn A\in\mathbb{R}^{nxn}, then for all v∈Rn,∣∣Av∣∣≤C∣∣A∣∣∣∣v∣∣v\in\mathbb{R}^n, ||Av|| \leq C ||A|| ||v|| . ∣∣v∣∣ ||v|| is defined to be the Euclidean norm for all vectors in R^n. And the norm of A is defined to be the maximum element of A (I think).
Original post by marcus2001
Let n∈Nn\in\mathbb{N}. Show that there exists a constant C (depending only n) such that if A∈Rnxn A\in\mathbb{R}^{nxn}, then for all v∈Rn,∣∣Av∣∣≤C∣∣A∣∣∣∣v∣∣v\in\mathbb{R}^n, ||Av|| \leq C ||A|| ||v|| . ∣∣v∣∣ ||v|| is defined to be the Euclidean norm for all vectors in R^n. And the norm of A is defined to be the maximum element of A (I think).


If it's the max norm, then it's:

∥A∥max=max⁡{∣aij∣}\|A\|_{\text{max}} = \max \{|a_{ij}|\}

Long time since I looked at this. That said, it may be useful if you can use:

∥Av∥2≤∥Av∥1\|Av\|_2\leq \|Av\|_1 which would then allow the expansion of ∥Av∥1\|Av\|_1

Depends what you've covered.

Can't offer more than that as too rusty, but there's plenty of more knowledgeable people than I on here.
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Smaug123
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Original post by marcus2001
Let n∈Nn\in\mathbb{N}. Show that there exists a constant C (depending only n) such that if A∈Rnxn A\in\mathbb{R}^{nxn}, then for all v∈Rn,∣∣Av∣∣≤C∣∣A∣∣∣∣v∣∣v\in\mathbb{R}^n, ||Av|| \leq C ||A|| ||v|| . ∣∣v∣∣ ||v|| is defined to be the Euclidean norm for all vectors in R^n. And the norm of A is defined to be the maximum element of A (I think).

By the way, you wanted \times in the superscript of the first \mathbb{R}.

The way we did it in my Analysis course was to show that the operator norm ∣∣A∣∣=sup⁡{Ax∣∣x∣=1}||A|| = \sup\{Ax \mid |x|=1 \} is a norm, and then to show that all norms on finite-dimensional spaces are Lipschitz equivalent. That is the required result. Can you do either of those independently?

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