Talk:Matrix norm

The following page will be replaced by a table.--wshun 01:34, 8 Aug 2003 (UTC)

The most "natural" of these operator norms is the one which arises from the Euclidean norms ||.||₂ on K^m and Kⁿ. It is unfortunately relatively difficult to compute; we have

<math>\|A\|_2=\mbox{ the largest singular value of } A<math>

(see singular value). If we use the taxicab norm ||.||₁ on K^m and Kⁿ, then we obtain the operator norm

<math>\|A\|_1=\max_{1\le j\le n} \sum_{i=1}^m |a_{ij}|<math>

and if we use the maximum norm ||.||_∞ on K^m and Kⁿ, we get

<math>\|A\|_\infty=\max_{1\le i\le m} \sum_{j=1}^n |a_{ij}|<math>

The following inequalities obtain among the various discussed matrix norms for the m-by-n matrix A:

<math>

\frac{1}{\sqrt{n}}\Vert\,A\,\Vert_\infty \leq \Vert\,A\,\Vert_2 \leq \sqrt{m}\Vert\,A\,\Vert_\infty <math>

<math>

\frac{1}{\sqrt{m}}\Vert\,A\,\Vert_1 \leq \Vert\,A\,\Vert_2 \leq \sqrt{n}\Vert\,A\,\Vert_1 <math>

<math>

\Vert\,A\,\Vert_2 \leq \Vert\,A\,\Vert_F\leq\sqrt{n}\Vert\,A\,\Vert_2 <math>

What's wrong with Frobenius norm?

Why does the article say that Frobenius norm is not sub-multiplicative? It does satisfy the condition <math>\|A B\|\leq \|A\| \|B\|<math>, which can be easily proved as follows: <math> \|A B\|^2_F = \sum_{i,j=1}^n |\sum_{k=1}^n a_{i,k} b_{k,j}|^2 \leq \sum_{i,j=1}^n \Big(\sum_{k=1}^n |a_{i,k}|^2\Big) \Big(\sum_{l=1}^n |b_{l,j}|^2\Big) = <math> <math>=(\sum_{i,k=1}^n |a_{i,k}|^2) (\sum_{j,l=1}^n |b_{l,j}|^2) = \|A\|^2_F \|B\|^2_F <math>. --Igor 21:21, Feb 18, 2005 (UTC)

What happened to the article?

The above discussion suggests that the article used to be more extensive. However, the revision history of the current article shows only one edit, by CyborgTosser on 25 Feb 2005. Did something drastic happen to the article? -- Jitse Niesen 11:36, 2 Mar 2005 (UTC)

I'm not quite sure what happened. Apparently there used to be an article here, but the content must have been moved. I'm not sure where and I'm not sure why, but a lot of articles link here, so I figured we needed the article. Hopefully whoever moved the content will replace whatever is relevant. CyborgTosser (Only half the battle) 03:21, 11 Mar 2005 (UTC)

I don't know either. I couldn't find the old page on wikipedia with google, but I've put a copy (from a wikipedia clone) at Matrix norm/old. Lupin 13:50, 11 Mar 2005 (UTC)

It seems that User:RickK deleted this page after it had been vandalised. Idiot. I've asked him to restore it with edit history to a subpage if possible. Lupin 14:10, 11 Mar 2005 (UTC)

Induced norm

I'm a little confused where the article says that "any induced norm satisfies the inequality ...". Is the intended meaning that the operator norm satisfies that inequality, or are there other norms which are also known as induced norms which satisfy that inequality? If the former, it should be rephrased as "the induced norm satisfies..." and if the latter, an explanation of what is meant by an induced norm should be given. Lupin 01:24, 11 May 2005 (UTC)

The terms "induced norm" and "operator norm" are synonymous. I used "any induced norm" instead of "the induced norm" because there are several operator norms. One example is the spectral norm, another example arises when one takes the ∞-norm on Kⁿ, defined by

<math> \|v\|_\infty = \max_i |v_i|; <math>

the resulting operator norm is

<math> \|A\|_\infty = \max_i \sum_j |a_{ij}|. <math>

I hope this resolves the confusion; feel free (of course) to edit the article to make it clearer. -- Jitse Niesen 10:23, 11 May 2005 (UTC)