Re: Orthogonazlie with Method->"Householder"
- To: mathgroup at smc.vnet.net
- Subject: [mg97815] Re: Orthogonazlie with Method->"Householder"
- From: Ray Koopman <koopman at sfu.ca>
- Date: Sun, 22 Mar 2009 05:49:42 -0500 (EST)
- References: <gpvh97$grj$1@smc.vnet.net>
On Mar 20, 12:40 am, Maris Ozols <maroz... at gmail.com> wrote: > I have a large numeric matrix M (see below), whose rank is 6. Clearly, > the rank of Orthogonalize[M] must be the same (in particular, it > should not increase). However, if I execute the following code on my > matrix M: > > MatrixRank@M > MatrixRank@Orthogonalize[M] > MatrixRank@Orthogonalize[M,Method->"Householder"] > > I get 6, 6, and 12. So it appears to me that Householder's method > has doubled the rank of M. I am using Mathematica 7.0 in Linux. > > Is this a bug or a feature? > > ~Maris Ozols~ > > Here is my matrix: > > M= << data deleted >> Are these the singular values you got? It sure looks to me like the rank is 6. In[1]:= Dimensions[M = << data deleted >> ] Out[1]= {21,64} In[2]:= SingularValueList[M,Tolerance->0] //InputForm Out[2]//InputForm= {44.44844462032234, 44.44844462032234, 27.042668434447457, 17.70074509587277, 8.904356780478404, 8.904356780478402, 8.017322620768825*^-15, 5.390291801559869*^-15, 3.217419606356955*^-15, 1.8374826839384085*^-15, 1.1177979027971086*^-15, 6.299097734357916*^-16, 2.435981051042932*^-16, 1.164217615077378*^-16, 9.350573374138264*^-17, 7.638745641833149*^-17, 8.251986281631452*^-31, 1.8830435203966325*^-31, 1.5994058233690583*^-32, 8.05780199579207*^-48, 0.} Delete the all-zero rows and columns. The nonzero singular values should not change. In[3]:= Dimensions[ m = Transpose@Select[Transpose@Select[M,#.#>0&],#.#>0&]] Out[3]= {18,43} In[4]:= SingularValueList[m,Tolerance->0] //InputForm Out[4]//InputForm= {44.44844462032235, 44.44844462032235, 27.042668434447467, 17.70074509587277, 8.904356780478398, 8.904356780478398, 9.333262724578447*^-15, 3.826650949746316*^-15, 3.1168179124846886*^-15, 1.944136451937067*^-15, 7.03860275960609*^-16, 2.880589245914001*^-16, 2.017780442593996*^-16, 1.7348598893568469*^-16, 1.2196907654448287*^-31, 5.311192383519828*^-32, 5.570074887646594*^-33, 8.074404526800988*^-49}