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equivalent to the domain being compact? In the case of the induced norm that would imply (from my perspective) max in the case abs(x)<=1 and supremum in the case x not equal to zero. I am not sure if it is actually an issue or not because at least in case of the induced 2 norm, the supremum is actually part of the range. That in turn implies to me that the supremum is reached for any similarly defined induced norm because of the equivalence of norms in finite dimensional spaces. Can someone with experience maybe point out the disconnect I seem to be having?
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Doesn't |A| specify the absolute value? Using the correct notation yields ||A||β€||A|| for all ||A||. Isn't that self evident? Furthermore m and n are not specified. Therefore I have removed this section till someone can clarify this content. It looks as if though someone partially moved content
1710:
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.
4940:
The clashing notations here are so confusing. I see people use ||T||_p for the
Schatten norms all the time, but I don't see this notation meaning something else. For the sake of having a readable article, I would suggest we use different notations for the other ones. Do other people think the other
4615:
It would be much clearer if the definitions of the norms and their properties was more clearly demarcated. At present, being sub-multiplicative is defined at the top, but the fact that all induced norms are sub-multiplicative is just mentioned in passing in the discussion of induced norms. Contrast
2826:
I'm interested in learning about the gradient of the matrix norm but I can't seem to find this information within wikipedia. I guess I'm requesting a new article and I don't know where to do that, but it seems logical for this article to point me to the gradient of the norm (maybe under see also).
2770:
article is not really clear about the equivalence of norms: since we are talking about matrices of finite size, all vector norms should be equivalent. the bunch of inequalities in the bottom could (mis)lead the reader into thinking otherwise. if, in addition, submultiplicativity is required, does
3946:
In some of the definitions I wasn't sure if max should actually be the supremum. I thought a maximum is guaranteed to exist for compact sets of real numbers, but not necessarily for open sets. In the case of linear, finite-dimensional operators(open sets are mapped to open sets) wouldn't this be
1754:
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
2166:
in two different norms. That is probably the reason why it is mentioned that the submultiplicative property holds for square matrices only. However, in the special case of the 2-norm the definition this is wrong. But even without the special case it is misleading for the reader, as the
4919:, which was used in other sections in this article), and I also changed the other parts of this section accordingly. One more thing: the inequality between the induced 2-norm and the Frobenius norm is mentioned before the Frobenius norm section, so probably we should change this.
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Start with the definition of a matrix norm, then go through the definitions of induced, Frobenius etc. norms as examples. Then go through the definitions of each property matrix norms might have, with clear results on which norms (do not) possess the given
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It took me two days time to figure out that the statement on
Knowledge (XXG) about submultiplicative property was misleading. As said, the submultiplicative property also holds for consistent p-norms, be it that in this case you are actually splitting
4623:
Start with the definition of a matrix norm, and the formal definition of each property that such a norm might have. Then go through the definitions of induced, Frobenius etc. norms, with clear results for each norm on which properties it does (not)
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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 β-norm on
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I came here looking for an introduction to the concept of matrix norms and an understanding of why they are important and what their applications are. The article lacks any of this information - it would be very useful to have here.
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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? --
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4311:{\displaystyle \|A\|={\sqrt {\lambda _{max}(A^{*}A)}}={\sqrt {\lambda _{max}((P^{-1})^{*}D^{*}P^{*}PDP^{-1})}}={\sqrt {\lambda _{max}((P^{-1})^{*}P^{*}D^{*}DPP^{-1})}}={\sqrt {\lambda _{max}(D^{*}D)}}}
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I removed the condition that the matrix be square for the induced norm (when p = 2) to be equivalent to the largest singular value. Indeed, this equivalence is true for non-square matrices too.
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as a matrix generalization of HΓΆlder's inequality. It turns out this was for
Schatten norm, not for induced p-norm. So I moved it to the Schatten norm section with a hint about how to derive it.
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1207:{\displaystyle \|AB\|_{F}^{2}=\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 )}=}
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was given for the
Frobenius norm, which only holds for real matrices (without any reference to this restirction). I now changed this, adding the correct definition (using the notation
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The
Frobenius norm is sub-multiplicative and is very useful for numerical linear algebra. The sub-multiplicativity of Frobenius norm can be proved using CauchyβSchwarz inequality.
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I feel that this article is quite unclear about when submultiplicativity applies. In particular, it should be made clear that for matrix norms based on vectors p-norms that for
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There is a statement in the article: "For a symmetric or hermitian matrix A, we have equality for the 2-norm, since in this case the 2-norm is the spectral radius of A"
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Matrix norm on the set of all nxn matrices is a real value function, ||.|| defined on this set, satisfying for all nxn matrices A and B and all real number
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You are right that this could be added. So, why don't you change the article to include this? You can edit the article by clicking on "edit this page", see
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is the
Euclidean norm which is the same as the Frobenius norm if the input vector is treated like a matrix, but when the input is a matrix, the notation
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103:
1545:. Also it is also called the Hilbert-Schmidt norm, because the page for Hilbert-Schmidt norm says that it is only analogous to the Frobenius norm.--
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It would be a useful improvement to this article if the meaning of this submultiplicativity were to be also stated in mathematical notation.
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This article was very useful. I was getting confused with that double-meaning notation and this article clarified it. Sorry for my
English.--
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620:{\displaystyle {\frac {1}{\sqrt {n}}}\Vert \,A\,\Vert _{\infty }\leq \Vert \,A\,\Vert _{2}\leq {\sqrt {m}}\Vert \,A\,\Vert _{\infty }}
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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deleted this page after it had been vandalised. Idiot. I've asked him to restore it with edit history to a subpage if possible.
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this to consistency, for which the fact that induced norms are consistent is mentioned next to the definition of consistency.
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2167:"submultiplicative" definition is used in a much wider range than a norm that only splits in two equal norms. See page 5 of
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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
5186:{\displaystyle \|A\|_{2}={\sqrt {\rho (A^{*}A)}}\leq {\sqrt {\|A^{*}A\|_{\infty }}}\leq {\sqrt {\|A\|_{1}\|A\|_{\infty }}}}
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for details. Don't worry about making mistakes; you will be corrected if necessary. I look forward to your contributions,
723:{\displaystyle {\frac {1}{\sqrt {m}}}\Vert \,A\,\Vert _{1}\leq \Vert \,A\,\Vert _{2}\leq {\sqrt {n}}\Vert \,A\,\Vert _{1}}
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it isn't true that the trace norm, sum(sigma), is <= the Frob. norm, sum(sigma^2); e.g. suppose all sigma<1.
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induced norm satisfies..." and if the latter, an explanation of what is meant by an induced norm should be given.
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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4517:{\displaystyle \lambda (A_{1}A_{2}\cdots A_{n})=\lambda (A_{\sigma (1)}A_{\sigma (2)}\cdots A_{\sigma (n)})}
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1403:{\displaystyle =(\sum _{i,k=1}^{n}|a_{i,k}|^{2})(\sum _{j,l=1}^{n}|b_{l,j}|^{2})=\|A\|_{F}^{2}\|B\|_{F}^{2}}
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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on
Knowledge (XXG). If you would like to participate, please visit the project page, where you can join
4685:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit
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1553:@KFrance, That is not true. The Frobenius norm is the Hilbert-Schmidt norm, but it is not the same as
854:
Why does the article say that
Frobenius norm is not sub-multiplicative? It does satisfy the condition
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https://web.archive.org/web/20160304053759/https://cs.uwaterloo.ca/~watrous/CS766/LectureNotes/02.pdf
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this change? (apparently so, the article seems to imply the Banach algebra topology is not unique.)
1937:
I hope this resolves the confusion; feel free (of course) to edit the article to make it clearer. --
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814:{\displaystyle \Vert \,A\,\Vert _{2}\leq \Vert \,A\,\Vert _{F}\leq {\sqrt {n}}\Vert \,A\,\Vert _{2}}
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I hope someone knowledgeable about this subject can add the appropriate inequality to the article.
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The most "natural" of these operator norms is the one which arises from the
Euclidean norms ||.||
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before doing mass systematic removals. This message is updated dynamically through the template
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The domain is usually a sphere. These are closed and bounded, and thus compact by Heine-Borel.
3556:{\displaystyle \left\|A\right\|_{\infty }=\max \limits _{1\leq i\leq m}\sum _{j=1}^{n}|a_{ij}|,}
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4862:{\displaystyle \|A\|_{\rm {F}}={\sqrt {\operatorname {trace} \left(A^{\textsf {T}}A\right)}},}
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3648:{\displaystyle {\begin{bmatrix}3&5&7\\2&-6&4\\0&2&8\\\end{bmatrix}}}
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3444:{\displaystyle \left\|A\right\|_{1}=\max \limits _{1\leq j\leq n}\sum _{i=1}^{m}|a_{ij}|,}
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4711:, "External links modified" talk page sections are no longer generated or monitored by
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If you found an error with any archives or the URLs themselves, you can fix them with
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usually denotes spectral norm, which is not the Frobenius norm. @Igor, that is true.
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This is an important inequality, so I think it should be re-included on this page.
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The following inequalities obtain among the various discussed matrix norms for the
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In either approach, a table of norms and properties might help the presentation.
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2244:| Β· | is a (submultiplicative) matrix norm. A matrix norm || Β· || is said to be
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is a special case of diagonalizable matrices when the diagonalizing matrix are
4717:. No special action is required regarding these talk page notices, other than
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if there exists no other matrix norm | Β· | satisfying |A|β€||A|| for all |A|.
4318:(since the set of eigenvalues of AB is same as the set of eigenvalue of BA)
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I guess the equality actually holds for more general case: It holds for any
500:{\displaystyle \|A\|_{\infty }=\max _{1\leq i\leq m}\sum _{j=1}^{n}|a_{ij}|}
2799:
The article doesn't explain why the "trace norm" is an "entry-wise norm".
2306:. More-over I will just add these statements back in and reword them. --
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All induced vector norms upper bound the spectral radius, in particular,
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Update: I edited the page. Can somebody check? Does it need references?
2236:, then for any vector norm | Β· |, there exists a unique positive number
380:{\displaystyle \|A\|_{1}=\max _{1\leq j\leq n}\sum _{i=1}^{m}|a_{ij}|}
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is the maximum absolute row sum of the matrix. In addition both
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214:. It is unfortunately relatively difficult to compute; we have
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So sorry; don't know what I was thinking. I will just change
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The two following functions are two examples of matrix norm:
2109:. This is shown in Proposition 2.7.2 on the following page
1926:{\displaystyle \|A\|_{\infty }=\max _{i}\sum _{j}|a_{ij}|.}
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The section on the Frobenius norm contains this sentence:
5042:{\displaystyle |A|_{F}\leq {\sqrt {|A|_{1}|A|_{\infty }}}}
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https://cs.uwaterloo.ca/~watrous/CS766/LectureNotes/02.pdf
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for additional information. I made the following changes:
2671:{\displaystyle \|A\|_{\infty }\leq {\sqrt {n}}\|A\|_{2}}
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is the maximum absolute column sum of the matrix, and
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1608:{\displaystyle \|A\|={\sqrt {\lambda _{max}A^{H}A}}}
1538:{\displaystyle \|A\|={\sqrt {\lambda _{max}A^{H}A}}}
98:, a collaborative effort to improve the coverage of
4941:kinds of norms take precedence for this notation?
4721:using the archive tool instructions below. Editors
4321:Does anybody see any problem with this argument? -
3740:{\displaystyle \left\|A\right\|_{1}=|7|+|4|+|8|=19}
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4619:I would suggest one of the following two layouts:
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132:This article has not yet received a rating on the
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4650:Article contains no motivations or applications
2298:refereed to. I was reading a book earlier and
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1988:{\displaystyle A\in {\mathbb {C} }^{m\times n}}
1483:norm that is mentioned earlier in the article.
4707:This message was posted before February 2018.
3932:are the special norm of a general norm called
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2439:{\displaystyle \|A\|_{1}\leq n\|A\|_{\infty }}
2749:{\displaystyle A\in \mathbb {R} ^{m\times n}}
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4677:I have just modified one external link on
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4965:The text as of 2013-03-29 claimed that
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4373:gives the set of eigenvalues of matrix
3197:{\displaystyle \|A+B\|\leq \|A\|+\|B\|}
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112:Knowledge (XXG):WikiProject Mathematics
49:
19:
4948:2607:9880:1A18:10A:64C9:2106:FDEB:3FFD
4670:External links modified (January 2018)
4013:, which in turn, is a special case of
2302:was refereed to as the determinant of
1417:Is it true that the Frobenius norm is
5239:Unknown-priority mathematics articles
4783:Frobenius norm - corrected definition
3324:{\displaystyle \|AB\|\leq \|A\|\|B\|}
2807:) 14:49, 23 July 2008 (UTC) Fixed. --
2708:{\displaystyle A\in \mathbb {R} ^{n}}
897:{\displaystyle \|AB\|\leq \|A\|\|B\|}
7:
3890:{\displaystyle \left\|A\right\|_{1}}
3820:{\displaystyle \left\|A\right\|_{1}}
2294:, it's clear from the sentence what
390:and if we use the maximum norm ||.||
92:This article is within the scope of
4020:Trivial proof: Let A = P D P. Then
282:, then we obtain the operator norm
270:). If we use the taxicab norm ||.||
38:It is of interest to the following
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831:http://de.wikipedia.org/Matrixnorm
612:
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14:
4681:. Please take a moment to review
4017:. All these are diagonalizable.)
2252:such that it's meaning was lost.
850:What's wrong with Frobenius norm?
4961:HΓΆlder's inequality for matrices
4590:Centralized discussion on proofs
2321:
2219:
829:This site needs to be linked to
115:Template:WikiProject Mathematics
79:
69:
51:
20:
4899:for the conjugate transpose of
1849:the resulting operator norm is
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4606:17:58, 29 September 2015 (UTC)
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3569:For examples: With matrix A:
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3273:{\displaystyle K^{m\times n}.}
3031:
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2208:14:50, 20 September 2016 (UTC)
2186:14:00, 20 September 2016 (UTC)
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4665:14:07, 22 December 2015 (UTC)
4645:20:56, 14 November 2015 (UTC)
3140:{\displaystyle K^{m\times n}}
2781:trace norm vs. Frobenius norm
2776:14:08, 13 February 2007 (UTC)
2761:03:38, 24 December 2006 (UTC)
2311:01:06, 29 December 2006 (UTC)
2272:02:53, 24 December 2006 (UTC)
1697:What happened to the article?
106:and see a list of open tasks.
5234:C-Class mathematics articles
5053:correction to the correction
4790:Previously, the definition
4775:15:43, 21 January 2018 (UTC)
4582:15:31, 19 October 2013 (UTC)
4562:04:23, 8 February 2013 (UTC)
4344:That argument was wrong. If
4339:18:47, 7 February 2013 (UTC)
4936:Horrible clashing notations
4931:12:19, 10 August 2019 (UTC)
4544:is a cyclic permutation. -
4366:{\displaystyle \lambda (A)}
3485:
3373:
2790:16:34, 8 October 2007 (UTC)
2316:Matrix Norm not Vector Norm
2130:11:24, 12 August 2005 (UTC)
845:19:03, 5 January 2011 (UTC)
5255:
4738:(last update: 5 June 2024)
4674:Hello fellow Wikipedians,
3985:17:06, 1 August 2011 (UTC)
2817:13:02, 27 April 2011 (UTC)
2217:
1719:) 03:21, 11 Mar 2005 (UTC)
1414:21:21, Feb 18, 2005 (UTC)
188:01:43, 16 April 2020 (UTC)
176:least upper bound property
4787:Dear fellow Wikipedians,
4596:WT:MATH#Proofs, revisited
3963:18:49, 19 July 2010 (UTC)
2906:{\displaystyle \|A\|: -->
1674:{\displaystyle \|A\|_{2}}
1641:{\displaystyle \|a\|_{2}}
1549:13:40, Oct 9, 2007 (MST)
1476:{\displaystyle \|A\|_{2}}
1443:{\displaystyle \|A\|_{p}}
131:
64:
46:
4956:06:58, 5 June 2021 (UTC)
1942:10:23, 11 May 2005 (UTC)
1764:01:24, 11 May 2005 (UTC)
1745:14:10, 11 Mar 2005 (UTC)
1732:13:50, 11 Mar 2005 (UTC)
1706:11:36, 2 Mar 2005 (UTC)
1691:17:54, 9 July 2014 (UTC)
199:01:34, 8 Aug 2003 (UTC)
165:00:08, 8 June 2015 (UTC)
134:project's priority scale
4537:{\displaystyle \sigma }
3792:: In the above example
3067:{\displaystyle \alpha }
2966:{\displaystyle \|A\|=0}
2934:{\displaystyle A\neq 0}
2870:{\displaystyle \alpha }
95:WikiProject Mathematics
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2853:Matrix Norm Definition
2758:ANONYMOUS COWARD0xC0DE
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2308:ANONYMOUS COWARD0xC0DE
2260:ANONYMOUS COWARD0xC0DE
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2159:{\displaystyle \|Ax\|}
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28:This article is rated
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4892:{\displaystyle A^{*}}
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32:on Knowledge (XXG)'s
5204:Submultiplicativity?
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2822:Gradient of the Norm
2766:equivalence of norms
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2715:and not of the form
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155:What is sup(x)Β ????
118:mathematics articles
4709:After February 2018
4611:Poor article layout
4007:symmetric/hermitian
3338:Matrix Norm Example
2992:{\displaystyle A=0}
1948:Submultiplicativity
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4763:InternetArchiveBot
4714:InternetArchiveBot
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3934:p-norm for vectors
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2118:How to edit a page
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87:Mathematics portal
34:content assessment
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4912:{\displaystyle A}
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4386:{\displaystyle A}
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3786:= |3|+|5|+|7|=15
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3237:{\displaystyle B}
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3204:for all matrices
3107:{\displaystyle A}
3094:and all matrices
3087:{\displaystyle K}
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2566:
2565:
2544:
2518:
2507:
2506:
2485:
2459:
2448:
2447:
2426:
2404:
2393:
2392:
2371:
2345:
2334:
2333:
2330:
2320:
2318:
2254:βThe preceding
2249:
2228:Moreover, when
2226:
2225:
2220:
2216:
2139:
2138:
2089:
2073:
2054:
2040:
2039:
2008:
1997:
1996:
1965:
1954:
1953:
1950:
1902:
1864:
1853:
1852:
1817:
1789:
1778:
1777:
1752:
1726:Matrix norm/old
1699:
1661:
1650:
1649:
1628:
1617:
1616:
1590:
1574:
1555:
1554:
1520:
1504:
1485:
1484:
1463:
1452:
1451:
1430:
1419:
1418:
1340:
1324:
1274:
1258:
1215:
1214:
1182:
1166:
1114:
1098:
1023:
1007:
991:
906:
905:
856:
855:
852:
827:
801:
771:
748:
733:
732:
710:
680:
657:
630:
629:
607:
577:
554:
527:
526:
479:
418:
407:
406:
393:
359:
298:
287:
286:
273:
230:
219:
218:
205:
153:
117:
114:
111:
108:
107:
85:
78:
29:
12:
11:
5:
5252:
5250:
5242:
5241:
5236:
5226:
5225:
5205:
5202:
5178:
5174:
5170:
5167:
5162:
5158:
5154:
5151:
5146:
5139:
5135:
5131:
5126:
5122:
5118:
5113:
5108:
5105:
5100:
5096:
5092:
5089:
5084:
5079:
5075:
5071:
5068:
5054:
5051:
5034:
5029:
5024:
5020:
5014:
5009:
5004:
5000:
4994:
4989:
4984:
4979:
4975:
4962:
4959:
4937:
4934:
4908:
4886:
4882:
4870:
4869:
4858:
4852:
4848:
4837:
4832:
4828:
4825:
4820:
4814:
4809:
4805:
4802:
4784:
4781:
4779:
4757:
4756:
4749:
4702:
4701:
4693:Added archive
4671:
4668:
4651:
4648:
4630:
4629:
4625:
4612:
4609:
4591:
4588:
4586:
4569:
4566:
4565:
4564:
4533:
4513:
4508:
4505:
4502:
4499:
4495:
4491:
4486:
4483:
4480:
4477:
4473:
4467:
4464:
4461:
4458:
4454:
4450:
4447:
4444:
4441:
4436:
4432:
4428:
4423:
4419:
4413:
4409:
4405:
4402:
4382:
4362:
4359:
4356:
4353:
4305:
4302:
4297:
4293:
4289:
4284:
4281:
4278:
4274:
4268:
4263:
4258:
4255:
4251:
4247:
4244:
4239:
4235:
4229:
4225:
4219:
4215:
4209:
4206:
4202:
4198:
4195:
4190:
4187:
4184:
4180:
4174:
4169:
4164:
4161:
4157:
4153:
4150:
4145:
4141:
4135:
4131:
4125:
4121:
4115:
4112:
4108:
4104:
4101:
4096:
4093:
4090:
4086:
4080:
4075:
4072:
4067:
4063:
4059:
4054:
4051:
4048:
4044:
4038:
4035:
4032:
4029:
4005:A. (Note that
4003:diagonalizable
3996:
3993:
3990:
3989:
3988:
3977:79.131.226.245
3955:79.235.159.125
3943:
3940:
3919:
3914:
3911:
3908:
3884:
3879:
3876:
3873:
3849:
3844:
3841:
3838:
3814:
3809:
3806:
3803:
3773:
3768:
3765:
3762:
3736:
3733:
3729:
3725:
3721:
3717:
3713:
3709:
3705:
3701:
3697:
3693:
3689:
3685:
3680:
3675:
3672:
3669:
3658:We have:
3657:
3642:
3636:
3633:
3631:
3628:
3626:
3623:
3622:
3619:
3616:
3614:
3611:
3608:
3606:
3603:
3602:
3599:
3596:
3594:
3591:
3589:
3586:
3585:
3583:
3572:
3566:
3564:
3563:
3552:
3548:
3542:
3539:
3535:
3530:
3524:
3519:
3516:
3513:
3509:
3503:
3500:
3497:
3494:
3491:
3487:
3483:
3478:
3473:
3470:
3467:
3440:
3436:
3430:
3427:
3423:
3418:
3412:
3407:
3404:
3401:
3397:
3391:
3388:
3385:
3382:
3379:
3375:
3371:
3366:
3361:
3358:
3355:
3339:
3336:
3332:
3331:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3280:
3269:
3264:
3261:
3258:
3254:
3233:
3213:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3147:
3134:
3131:
3128:
3124:
3103:
3083:
3063:
3043:
3040:
3037:
3033:
3029:
3025:
3021:
3018:
3015:
3012:
3009:
2999:
2988:
2985:
2982:
2962:
2959:
2956:
2953:
2950:
2930:
2927:
2924:
2902:
2899:
2896:
2893:
2890:
2866:
2854:
2851:
2823:
2820:
2796:
2793:
2782:
2779:
2767:
2764:
2743:
2740:
2737:
2732:
2727:
2724:
2702:
2697:
2692:
2689:
2665:
2661:
2657:
2654:
2649:
2644:
2639:
2635:
2631:
2628:
2606:
2602:
2598:
2595:
2592:
2589:
2584:
2580:
2576:
2573:
2551:
2547:
2543:
2540:
2535:
2530:
2525:
2521:
2517:
2514:
2492:
2488:
2484:
2481:
2476:
2471:
2466:
2462:
2458:
2455:
2433:
2429:
2425:
2422:
2419:
2416:
2411:
2407:
2403:
2400:
2378:
2374:
2370:
2367:
2362:
2357:
2352:
2348:
2344:
2341:
2331:
2319:
2317:
2314:
2227:
2218:
2215:
2212:
2200:94.210.213.220
2190:
2189:
2178:94.210.213.220
2155:
2152:
2149:
2146:
2133:
2132:
2096:
2092:
2088:
2085:
2080:
2076:
2072:
2069:
2066:
2061:
2057:
2053:
2050:
2047:
2025:
2022:
2019:
2013:
2007:
2004:
1982:
1979:
1976:
1970:
1964:
1961:
1949:
1946:
1945:
1944:
1935:
1934:
1933:
1922:
1918:
1912:
1909:
1905:
1900:
1894:
1890:
1884:
1880:
1876:
1871:
1867:
1863:
1860:
1847:
1846:
1845:
1834:
1830:
1824:
1820:
1815:
1809:
1805:
1801:
1796:
1792:
1788:
1785:
1755:rephrased as "
1751:
1748:
1747:
1746:
1737:It seems that
1734:
1733:
1721:
1720:
1698:
1695:
1694:
1693:
1668:
1664:
1660:
1657:
1635:
1631:
1627:
1624:
1602:
1597:
1593:
1587:
1584:
1581:
1577:
1571:
1568:
1565:
1562:
1532:
1527:
1523:
1517:
1514:
1511:
1507:
1501:
1498:
1495:
1492:
1470:
1466:
1462:
1459:
1437:
1433:
1429:
1426:
1397:
1392:
1388:
1384:
1381:
1376:
1371:
1367:
1363:
1360:
1357:
1354:
1349:
1344:
1337:
1334:
1331:
1327:
1322:
1316:
1311:
1308:
1305:
1302:
1299:
1295:
1291:
1288:
1283:
1278:
1271:
1268:
1265:
1261:
1256:
1250:
1245:
1242:
1239:
1236:
1233:
1229:
1225:
1222:
1203:
1198:
1191:
1186:
1179:
1176:
1173:
1169:
1164:
1158:
1153:
1150:
1147:
1143:
1137:
1130:
1123:
1118:
1111:
1108:
1105:
1101:
1096:
1090:
1085:
1082:
1079:
1075:
1069:
1062:
1057:
1054:
1051:
1048:
1045:
1041:
1037:
1032:
1027:
1020:
1017:
1014:
1010:
1004:
1001:
998:
994:
988:
983:
980:
977:
973:
968:
962:
957:
954:
951:
948:
945:
941:
937:
932:
927:
923:
919:
916:
913:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
851:
848:
826:
823:
822:
821:
808:
804:
798:
793:
788:
783:
778:
774:
768:
763:
760:
755:
751:
745:
740:
730:
717:
713:
707:
702:
697:
692:
687:
683:
677:
672:
669:
664:
660:
654:
649:
643:
639:
627:
614:
610:
604:
599:
594:
589:
584:
580:
574:
569:
566:
561:
557:
551:
546:
540:
536:
508:
507:
495:
489:
486:
482:
477:
471:
466:
463:
460:
456:
450:
447:
444:
441:
438:
434:
430:
425:
421:
417:
414:
391:
388:
387:
375:
369:
366:
362:
357:
351:
346:
343:
340:
336:
330:
327:
324:
321:
318:
314:
310:
305:
301:
297:
294:
271:
268:singular value
264:
263:
252:
242:
237:
233:
229:
226:
203:
152:
149:
146:
145:
142:
141:
138:
137:
130:
124:
123:
121:
104:the discussion
91:
90:
74:
62:
61:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
5251:
5240:
5237:
5235:
5232:
5231:
5229:
5222:
5219:
5216:
5214:
5209:
5203:
5201:
5200:
5199:User:Jfessler
5196:
5193:
5168:
5160:
5152:
5144:
5129:
5124:
5120:
5111:
5103:
5098:
5094:
5087:
5082:
5077:
5069:
5058:
5052:
5050:
5022:
5012:
5002:
4992:
4987:
4977:
4958:
4957:
4953:
4949:
4945:
4942:
4935:
4933:
4932:
4928:
4924:
4920:
4906:
4884:
4880:
4856:
4850:
4846:
4835:
4830:
4826:
4823:
4818:
4803:
4793:
4792:
4791:
4788:
4782:
4780:
4777:
4776:
4771:
4766:
4765:
4754:
4750:
4747:
4743:
4742:
4741:
4734:
4728:
4724:
4720:
4716:
4710:
4705:
4700:
4696:
4692:
4691:
4690:
4688:
4684:
4680:
4675:
4669:
4667:
4666:
4662:
4658:
4657:36.53.254.212
4649:
4647:
4646:
4642:
4638:
4633:
4626:
4622:
4621:
4620:
4617:
4610:
4608:
4607:
4604:
4601:
4597:
4589:
4587:
4584:
4583:
4579:
4575:
4574:147.83.79.107
4567:
4563:
4558:
4554:
4549:
4548:
4531:
4503:
4497:
4493:
4489:
4481:
4475:
4471:
4462:
4456:
4452:
4445:
4442:
4434:
4430:
4426:
4421:
4417:
4411:
4407:
4400:
4380:
4357:
4351:
4343:
4342:
4341:
4340:
4335:
4331:
4326:
4325:
4319:
4300:
4295:
4291:
4282:
4279:
4276:
4272:
4266:
4256:
4253:
4249:
4245:
4242:
4237:
4233:
4227:
4223:
4217:
4207:
4204:
4200:
4188:
4185:
4182:
4178:
4172:
4162:
4159:
4155:
4151:
4148:
4143:
4139:
4133:
4129:
4123:
4113:
4110:
4106:
4094:
4091:
4088:
4084:
4078:
4070:
4065:
4061:
4052:
4049:
4046:
4042:
4036:
4030:
4018:
4016:
4012:
4008:
4004:
3999:
3991:
3986:
3982:
3978:
3974:
3968:
3967:
3966:
3964:
3960:
3956:
3952:
3941:
3939:
3936:
3935:
3909:
3882:
3874:
3839:
3812:
3804:
3791:
3787:
3763:
3750:
3747:
3734:
3731:
3723:
3715:
3707:
3699:
3691:
3683:
3678:
3670:
3655:
3640:
3634:
3629:
3624:
3617:
3612:
3609:
3604:
3597:
3592:
3587:
3581:
3570:
3567:
3550:
3540:
3537:
3533:
3522:
3517:
3514:
3511:
3507:
3501:
3498:
3495:
3492:
3489:
3481:
3468:
3456:
3455:
3454:
3451:
3438:
3428:
3425:
3421:
3410:
3405:
3402:
3399:
3395:
3389:
3386:
3383:
3380:
3377:
3369:
3364:
3356:
3343:
3337:
3335:
3315:
3306:
3300:
3294:
3291:
3281:
3267:
3262:
3259:
3256:
3252:
3231:
3211:
3188:
3182:
3176:
3170:
3164:
3161:
3158:
3148:
3132:
3129:
3126:
3122:
3101:
3081:
3061:
3038:
3027:
3019:
3013:
3010:
3000:
2986:
2983:
2980:
2960:
2957:
2951:
2928:
2925:
2922:
2900:
2897:
2891:
2880:
2879:
2878:
2864:
2852:
2850:
2848:
2844:
2840:
2836:
2832:
2821:
2819:
2818:
2814:
2810:
2806:
2802:
2794:
2792:
2791:
2788:
2780:
2778:
2777:
2774:
2765:
2763:
2762:
2759:
2741:
2738:
2735:
2725:
2722:
2700:
2690:
2687:
2663:
2655:
2647:
2642:
2629:
2604:
2596:
2590:
2587:
2574:
2541:
2533:
2528:
2523:
2515:
2490:
2482:
2474:
2469:
2464:
2456:
2423:
2417:
2414:
2409:
2401:
2376:
2368:
2360:
2355:
2350:
2342:
2328:
2324:
2315:
2313:
2312:
2309:
2305:
2301:
2297:
2293:
2289:
2285:
2281:
2276:
2273:
2269:
2265:
2261:
2257:
2247:
2243:
2239:
2235:
2231:
2213:
2211:
2209:
2205:
2201:
2197:
2187:
2183:
2179:
2175:
2169:
2150:
2147:
2135:
2134:
2131:
2127:
2123:
2119:
2115:
2114:
2113:
2111:
2094:
2086:
2078:
2070:
2064:
2059:
2051:
2048:
2023:
2020:
2017:
2005:
2002:
1980:
1977:
1974:
1962:
1959:
1947:
1943:
1940:
1936:
1920:
1910:
1907:
1903:
1892:
1888:
1882:
1874:
1861:
1851:
1850:
1848:
1832:
1822:
1818:
1807:
1799:
1786:
1776:
1775:
1774:, defined by
1773:
1768:
1767:
1766:
1765:
1762:
1758:
1749:
1744:
1740:
1736:
1735:
1731:
1727:
1723:
1722:
1718:
1714:
1709:
1708:
1707:
1705:
1696:
1692:
1688:
1684:
1666:
1658:
1633:
1625:
1600:
1595:
1591:
1585:
1582:
1579:
1575:
1569:
1563:
1552:
1551:
1550:
1548:
1530:
1525:
1521:
1515:
1512:
1509:
1505:
1499:
1493:
1468:
1460:
1435:
1427:
1415:
1413:
1395:
1390:
1382:
1374:
1369:
1361:
1355:
1347:
1335:
1332:
1329:
1325:
1314:
1309:
1306:
1303:
1300:
1297:
1293:
1281:
1269:
1266:
1263:
1259:
1248:
1243:
1240:
1237:
1234:
1231:
1227:
1220:
1201:
1189:
1177:
1174:
1171:
1167:
1156:
1151:
1148:
1145:
1141:
1121:
1109:
1106:
1103:
1099:
1088:
1083:
1080:
1077:
1073:
1060:
1055:
1052:
1049:
1046:
1043:
1039:
1035:
1030:
1018:
1015:
1012:
1008:
1002:
999:
996:
992:
986:
981:
978:
975:
971:
960:
955:
952:
949:
946:
943:
939:
935:
930:
925:
917:
914:
888:
879:
873:
867:
864:
847:
846:
842:
838:
837:91.113.18.247
833:
832:
824:
806:
796:
786:
781:
776:
766:
758:
753:
743:
731:
715:
705:
695:
690:
685:
675:
667:
662:
652:
641:
637:
628:
602:
592:
587:
582:
572:
564:
549:
538:
534:
525:
524:
523:
521:
517:
513:
487:
484:
480:
469:
464:
461:
458:
454:
448:
445:
442:
439:
436:
428:
415:
405:
404:
403:
401:
397:
367:
364:
360:
349:
344:
341:
338:
334:
328:
325:
322:
319:
316:
308:
303:
295:
285:
284:
283:
281:
277:
269:
250:
240:
235:
227:
217:
216:
215:
213:
209:
200:
198:
193:
190:
189:
185:
181:
177:
173:
167:
166:
162:
158:
150:
135:
129:
126:
125:
122:
105:
101:
97:
96:
88:
82:
77:
75:
72:
68:
67:
63:
60:
57:
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50:
45:
41:
35:
27:
23:
18:
17:
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5212:
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5197:
5194:
5059:
5056:
4964:
4946:
4943:
4939:
4921:
4871:
4789:
4786:
4778:
4761:
4758:
4733:source check
4712:
4706:
4703:
4676:
4673:
4653:
4634:
4631:
4618:
4614:
4600:Arthur Rubin
4593:
4585:
4571:
4546:
4323:
4320:
4019:
4000:
3995:
3971:β Preceding
3945:
3937:
3788:
3751:
3748:
3656:
3571:
3568:
3565:
3452:
3344:
3341:
3333:
2856:
2825:
2798:
2784:
2769:
2679:
2303:
2299:
2295:
2291:
2287:
2283:
2279:
2277:
2250:
2245:
2241:
2237:
2233:
2229:
2214:Bad Notation
2194:βΒ Preceding
2191:
2172:βΒ Preceding
2122:Jitse Niesen
1951:
1939:Jitse Niesen
1771:
1756:
1753:
1750:Induced norm
1713:CyborgTosser
1704:Jitse Niesen
1700:
1416:
853:
834:
828:
519:
515:
511:
509:
399:
395:
389:
279:
275:
265:
211:
207:
201:
194:
191:
171:
168:
157:Dr. Universe
154:
93:
40:WikiProjects
4679:Matrix norm
3949:βPreceding
2835:Arbitrary18
2829:βPreceding
825:German Link
109:Mathematics
100:mathematics
59:Mathematics
5228:Categories
4770:Report bug
2913:0}" /: -->
2795:trace norm
2240:such that
1739:User:RickK
4753:this tool
4746:this tool
4628:property.
4568:Thank you
2787:Lpwithers
402:, we get
172:supremium
4923:Zimboras
4759:Cheers.β
4624:possess.
4557:contribs
4334:contribs
3973:unsigned
3951:unsigned
3054:for all
2884:0}": -->
2843:contribs
2831:unsigned
2327:Resolved
2268:contribs
2256:unsigned
2224:Resolved
2196:unsigned
2174:unsigned
4683:my edit
4393:, then
4011:unitary
2809:sattath
2801:sattath
2773:Mct mht
2292:||A||_p
2284:||A||_q
2246:minimal
1547:kfrance
518:matrix
174:or the
30:C-class
4944:Sam W
4603:(talk)
4547:Subh83
4324:Subh83
1683:Lavaka
178:of x.
36:scale.
4824:trace
3938:@@@@
3749:And:
3453:and
3334:@@@@
2898:: -->
2756:. --
2288:||A||
2038:that
1761:Lupin
1743:Lupin
1730:Lupin
266:(see
197:wshun
169:: -->
4952:talk
4927:talk
4661:talk
4641:talk
4594:See
4578:talk
4553:talk
4330:talk
3981:talk
3959:talk
3942:max?
3897:and
3790:Note
3224:and
2941:and
2839:talk
2813:talk
2805:talk
2286:and
2264:talk
2204:talk
2182:talk
2126:talk
1995:and
1687:talk
1412:Igor
1410:. --
841:talk
514:-by-
398:and
278:and
210:and
206:on
184:talk
161:talk
4727:RfC
4697:to
4637:cfp
4524:if
3486:max
3374:max
3244:in
3114:in
3074:in
2915:if
2300:|A|
2296:|A|
2290:to
2282:to
2280:|A|
2170:.
1879:max
1804:max
1757:the
433:max
394:on
313:max
274:on
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1921:.
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