Norm
in [Mathematica]
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From: Clausenator on 19 Jul 2006 05:36 Hi, I want to calculate a distance matrix, similar to (as poorly explained at) http://en.wikipedia.org/wiki/Distance_matrix I found out about the Function "Norm" in mathematica 5. Here is a little example. I want to calculate the distance between vectors {0,1} and {5,1}. The distance should be 5 Now, Norm[{{0., 1.}, {5., 1.}}, 2] results 5.10293 Norm[{{0., 1.} - {5., 1.}}, 2] results 5.0 According to the documentation I have (Mathematica Help Browser, search for "Norm" under "Built-in Functions") the version with the comma is documented. I like the solution with the dash better. Which one is it? In other words, is there some Wolfram description or can you explain the difference? Thanks for your help, Claus
From: Jens-Peer Kuska on 20 Jul 2006 06:27 Hi, and why do you think, that Norm[{{0., 1.}, {5., 1.}}, 2] which computes the maximum singular value of {{0., 1.}, {5., 1.}} has something to do with Norm[{{0., 1.} - {5., 1.}}, 2] which compute the Euclidian distance ?? And all this stand in the Help-Browser and in the reference of The Mathematica book ... Regards Jens <Clausenator(a)gmail.com> schrieb im Newsbeitrag news:e9kuel$l3h$1(a)smc.vnet.net... | Hi, | I want to calculate a distance matrix, similar to (as poorly explained | at) http://en.wikipedia.org/wiki/Distance_matrix | | I found out about the Function "Norm" in mathematica 5. | | Here is a little example. I want to calculate the distance between | vectors {0,1} and {5,1}. The distance should be 5 | | Now, | | Norm[{{0., 1.}, {5., 1.}}, 2] | results 5.10293 | | Norm[{{0., 1.} - {5., 1.}}, 2] | results 5.0 | | According to the documentation I have (Mathematica Help Browser, search | for "Norm" under "Built-in Functions") the version with the comma is | documented. I like the solution with the dash better. | Which one is it? In other words, is there some Wolfram description or | can you explain the difference? | | Thanks for your help, | Claus |
From: Carl K. Woll on 20 Jul 2006 06:33 Clausenator(a)gmail.com wrote: > Hi, > I want to calculate a distance matrix, similar to (as poorly explained > at) http://en.wikipedia.org/wiki/Distance_matrix > > I found out about the Function "Norm" in mathematica 5. > > Here is a little example. I want to calculate the distance between > vectors {0,1} and {5,1}. The distance should be 5 > > Now, > > Norm[{{0., 1.}, {5., 1.}}, 2] > results 5.10293 > Here you are computing the 2-norm of a matrix. > Norm[{{0., 1.} - {5., 1.}}, 2] > results 5.0 > Here you are computing the 2-norm of a vector. The norms of a matrix and a vector are not the same thing. Since you are interested in the distance between two points, you want to compute the magnitude of the difference of the two points, so you want to evaluate the norm of point1 minus point2. Carl Woll Wolfram Research > According to the documentation I have (Mathematica Help Browser, search > for "Norm" under "Built-in Functions") the version with the comma is > documented. I like the solution with the dash better. > Which one is it? In other words, is there some Wolfram description or > can you explain the difference? > > Thanks for your help, > Claus
From: Murray Eisenberg on 20 Jul 2006 06:37 Mathematically, the norm of a vector gives that vector's length. And the distance between two vectors is the norm of the difference between the two vectors. (What you call the "dash" is in fact a subtraction sign.) So, assuming you want the ordinary (that is, Euclidean) distance, the desired result is given by Norm[{0, 1, 5, 1}] and the result (in InputForm) is 3 Sqrt[3]. The final argument, 2, is superfluous in the case of the ordinary (Euclidean) norm, which is the 2-norm. It would help when doing such things if you were familiar, first, with the underlying mathematical ideas and second, with the documentation that Mathematica itself provides. For the latter, just evaluate ?Norm and then to get further information click the hyperlink in the output produced (or in the first instance look up Norm directly in the HelpBrowser). Clausenator(a)gmail.com wrote: > Hi, > I want to calculate a distance matrix, similar to (as poorly explained > at) http://en.wikipedia.org/wiki/Distance_matrix > > I found out about the Function "Norm" in mathematica 5. > > Here is a little example. I want to calculate the distance between > vectors {0,1} and {5,1}. The distance should be 5 > > Now, > > Norm[{{0., 1.}, {5., 1.}}, 2] > results 5.10293 > > Norm[{{0., 1.} - {5., 1.}}, 2] > results 5.0 > > According to the documentation I have (Mathematica Help Browser, search > for "Norm" under "Built-in Functions") the version with the comma is > documented. I like the solution with the dash better. > Which one is it? In other words, is there some Wolfram description or > can you explain the difference? > > Thanks for your help, > Claus > > -- Murray Eisenberg murray(a)math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305
From: Bill Rowe on 20 Jul 2006 06:38
On 7/19/06 at 5:21 AM, Clausenator(a)gmail.com wrote: >Hi, I want to calculate a distance matrix, similar to (as poorly >explained at) http://en.wikipedia.org/wiki/Distance_matrix >I found out about the Function "Norm" in mathematica 5. >Here is a little example. I want to calculate the distance between >vectors {0,1} and {5,1}. The distance should be 5 >Now, >Norm[{{0., 1.}, {5., 1.}}, 2] results 5.10293 >Norm[{{0., 1.} - {5., 1.}}, 2] results 5.0 >According to the documentation I have (Mathematica Help Browser, >search for "Norm" under "Built-in Functions") the version with the >comma is documented. I like the solution with the dash better. Which >one is it? In other words, is there some Wolfram description or can >you explain the difference? Yes, In[10]:= {{0.,1.},{5.,1.}}!={{0.,1.}-{5.,1.}} Out[10]= True Norm[{{0,.1,}-{5.,1.}},2] is exactly the same as Norm[{{5.,0}},2] which is 5. The dash tells Mathematica to do a subtraction then compute the norm. Norm[{{0.,1.},{5.,1.}},2] is the norm of a 2x2 matrix and is not equal to 5. In particular for a matrix, m, Norm[m] is a singular value of m. Also, the default for the Norm function is the 2-norm. That is In[4]:= Norm[{{0,1},{5,1}}]==Norm[{{0,1},{5,1}},2] Out[4]= True and In[9]:= Norm[{{0,1},{5,1}}//N]==First(a)SingularValueList[{{0,1},{5,1}}//N] Out[9]= True Finally, all of this is documented and can be found using the Help Browser. -- To reply via email subtract one hundred and four |