Orthogonazlie with Method->"Householder"
- To: mathgroup at smc.vnet.net
- Subject: [mg97744] Orthogonazlie with Method->"Householder"
- From: Maris Ozols <marozols at gmail.com>
- Date: Fri, 20 Mar 2009 02:40:33 -0500 (EST)
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= {{-1.367128300941737`,0,0,0,-5.917767019974907`,0,0,0,1.2566212950683242`,0= ,0,0,0.7325966416414449`,0,0,0,-5.917767019974907`,0,0,0,1.1371276109150097= `,0,0,0,0,0,0,0,0,0,0,0,1.2566212950683242`,0,0,0,0,0,0,0,-0.38052787110636= 51`,0,0,0,0,0,0,0,0.7325966416414449`,0,0,0,0,0,0,0,0,0,0,0,1.0344867092261= 78`,0,0,0}, {-1.367128300941737`,-5.917767019974907`,1.2566212950683242`,0.732596641641= 4449`,0,0,0,0,0,0,0,0,0,0,0,0,-5.917767019974907`,1.1371276109150097`,0,0,0= ,0,0,0,0,0,0,0,0,0,0,0,1.2566212950683242`,0,-0.3805278711063651`,0,0,0,0,0= ,0,0,0,0,0,0,0,0,0.7325966416414449`,0,0,1.0344867092261778`,0,0,0,0,0,0,0,= 0,0,0,0,0}, {-1.367128300941737`,0,0,0,-5.917767019974907`,0,0,0,1.2566212950683242`,0,= 0,0,0.7325966416414449`,0,0,0,-5.917767019974907`,0,0,0,1.1371276109150097`= ,0,0,0,0,0,0,0,0,0,0,0,1.2566212950683242`,0,0,0,0,0,0,0,-0.380527871106365= 1`,0,0,0,0,0,0,0,0.7325966416414449`,0,0,0,0,0,0,0,0,0,0,0,1.03448670922617= 8`,0,0,0}, {-1.367128300941737`,-5.917767019974907`,1.2566212950683242`,0.732596641641= 4449`,0,0,0,0,0,0,0,0,0,0,0,0,-5.917767019974907`,1.1371276109150097`,0,0,0= ,0,0,0,0,0,0,0,0,0,0,0,1.2566212950683242`,0,-0.3805278711063651`,0,0,0,0,0= ,0,0,0,0,0,0,0,0,0.7325966416414449`,0,0,1.0344867092261778`,0,0,0,0,0,0,0,= 0,0,0,0,0}, {-1.367128300941737`,-5.917767019974907`,1.2566212950683242`,0.732596641641= 4449`,-5.917767019974907`,1.1371276109150097`,0,0,1.2566212950683242`,0,-0.= 3805278711063651`,0,0.7325966416414449`,0,0,1.034486709226178`,0,0,0,0,0,0,= 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0= ,0,0,0,0}, {-1.367128300941737`,-5.917767019974907`,1.2566212950683242`,0.732596641641= 4449`,-5.917767019974907`,1.1371276109150097`,0,0,1.2566212950683242`,0,-0.= 3805278711063651`,0,0.7325966416414449`,0,0,1.034486709226178`,0,0,0,0,0,0,= 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0= ,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-0.557546880846985`,-2.599916056557537= `,0,0,0,0,0.557546880846985`,0,0,0.7873020502993349`,2.599916056557537`,0,-= 0.7873020502993349`,0,0,-1.6661117377481922`,0,-12.243702660922095`,1.66611= 17377481922`,0,0,2.352686800371387`,0,0,0,0,12.243702660922095`,-2.35268680= 0371387`,0,0,0,2.857877542171938`,-4.50375057162902`,0,-2.857877542171938`,= 0,0.8654174979155087`,0,4.50375057162902`,-0.8654174979155087`,0,0,0,0,0,0}= , {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,= 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-0.557546880846985`,-2.599916056557537= `,0,0,0,0,0.557546880846985`,0,0,0.7873020502993349`,2.599916056557537`,0,-= 0.7873020502993349`,0,0,-1.6661117377481922`,0,-12.243702660922095`,1.66611= 17377481922`,0,0,2.352686800371387`,0,0,0,0,12.243702660922095`,-2.35268680= 0371387`,0,0,0,2.857877542171938`,-4.50375057162902`,0,-2.857877542171938`,= 0,0.8654174979155087`,0,4.50375057162902`,-0.8654174979155087`,0,0,0,0,0,0}= , {0,0,0,0,0,0,-0.557546880846985`,-2.599916056557537`,0,-1.6661117377481922`= ,0,-12.243702660922095`,0,2.857877542171938`,-4.50375057162902`,0,0,0,0,0,0= ,0,0,0,1.6661117377481922`,0,0,2.352686800371387`,-2.857877542171938`,0,0.8= 654174979155087`,0,0,0,0,0,0.557546880846985`,0,0,0.7873020502993349`,0,0,0= ,0,4.50375057162902`,-0.8654174979155087`,0,0,0,0,0,0,2.599916056557537`,0,= -0.7873020502993349`,0,12.243702660922095`,-2.352686800371387`,0,0,0,0,0,0}= , {0,0,0,0,0,0,-0.557546880846985`,-2.599916056557537`,0,-1.6661117377481922`= ,0,-12.243702660922095`,0,2.857877542171938`,-4.50375057162902`,0,0,0,0,0,0= ,0,0,0,1.6661117377481922`,0,0,2.352686800371387`,-2.857877542171938`,0,0.8= 654174979155087`,0,0,0,0,0,0.557546880846985`,0,0,0.7873020502993349`,0,0,0= ,0,4.50375057162902`,-0.8654174979155087`,0,0,0,0,0,0,2.599916056557537`,0,= -0.7873020502993349`,0,12.243702660922095`,-2.352686800371387`,0,0,0,0,0,0}= , {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.557546880846985`,2.599916056557537`,= 0,0,0,0,-0.557546880846985`,0,0,-0.7873020502993349`,-2.599916056557537`,0,= 0.7873020502993349`,0,0,1.6661117377481922`,0,12.243702660922095`,-1.666111= 7377481922`,0,0,-2.352686800371387`,0,0,0,0,-12.243702660922095`,2.35268680= 0371387`,0,0,0,-2.857877542171938`,4.50375057162902`,0,2.857877542171938`,0= ,-0.8654174979155087`,0,-4.50375057162902`,0.8654174979155087`,0,0,0,0,0,0}= , {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,= 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,-1.6661117377481922`,2.857877542171938`,0,-0.557546880846985`,= 0,-4.50375057162902`,0,-2.599916056557537`,-12.243702660922095`,0,0,0,1.666= 1117377481922`,-2.857877542171938`,0,0,0,0,0,0,0,0.8654174979155087`,0,0,2.= 352686800371387`,0,0,0.557546880846985`,0,4.50375057162902`,0,0,0,-0.865417= 4979155087`,0,0,0,0,0,0.7873020502993349`,0,0,0,2.599916056557537`,12.24370= 2660922095`,0,0,0,-2.352686800371387`,0,0,-0.7873020502993349`,0,0,0,0,0,0}= , {0,0,0,0,0,0,-1.6661117377481922`,2.857877542171938`,0,-0.557546880846985`,= 0,-4.50375057162902`,0,-2.599916056557537`,-12.243702660922095`,0,0,0,1.666= 1117377481922`,-2.857877542171938`,0,0,0,0,0,0,0,0.8654174979155087`,0,0,2.= 352686800371387`,0,0,0.557546880846985`,0,4.50375057162902`,0,0,0,-0.865417= 4979155087`,0,0,0,0,0,0.7873020502993349`,0,0,0,2.599916056557537`,12.24370= 2660922095`,0,0,0,-2.352686800371387`,0,0,-0.7873020502993349`,0,0,0,0,0,0}= , {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-0.557546880846985`,-2.599916056557537= `,0,0,0,0,0.557546880846985`,0,0,0.7873020502993349`,2.599916056557537`,0,-= 0.7873020502993349`,0,0,-1.6661117377481922`,0,-12.243702660922095`,1.66611= 17377481922`,0,0,2.352686800371387`,0,0,0,0,12.243702660922095`,-2.35268680= 0371387`,0,0,0,2.857877542171938`,-4.50375057162902`,0,-2.857877542171938`,= 0,0.8654174979155087`,0,4.50375057162902`,-0.8654174979155087`,0,0,0,0,0,0}= , {0,0,0,0,0,0,-0.557546880846985`,-2.599916056557537`,0,-1.6661117377481922`= ,0,-12.243702660922095`,0,2.857877542171938`,-4.50375057162902`,0,0,0,0,0,0= ,0,0,0,1.6661117377481922`,0,0,2.352686800371387`,-2.857877542171938`,0,0.8= 654174979155087`,0,0,0,0,0,0.557546880846985`,0,0,0.7873020502993349`,0,0,0= ,0,4.50375057162902`,-0.8654174979155087`,0,0,0,0,0,0,2.599916056557537`,0,= -0.7873020502993349`,0,12.243702660922095`,-2.352686800371387`,0,0,0,0,0,0}= , {0,0,0,0,0,0,-0.557546880846985`,-2.599916056557537`,0,-1.6661117377481922`= ,0,-12.243702660922095`,0,2.857877542171938`,-4.50375057162902`,0,0,0,0,0,0= ,0,0,0,1.6661117377481922`,0,0,2.352686800371387`,-2.857877542171938`,0,0.8= 654174979155087`,0,0,0,0,0,0.557546880846985`,0,0,0.7873020502993349`,0,0,0= ,0,4.50375057162902`,-0.8654174979155087`,0,0,0,0,0,0,2.599916056557537`,0,= -0.7873020502993349`,0,12.243702660922095`,-2.352686800371387`,0,0,0,0,0,0}= , {0,0,0,0,0,0,-1.6661117377481922`,2.857877542171938`,0,-0.557546880846985`,= 0,-4.50375057162902`,0,-2.599916056557537`,-12.243702660922095`,0,0,0,1.666= 1117377481922`,-2.857877542171938`,0,0,0,0,0,0,0,0.8654174979155087`,0,0,2.= 352686800371387`,0,0,0.557546880846985`,0,4.50375057162902`,0,0,0,-0.865417= 4979155087`,0,0,0,0,0,0.7873020502993349`,0,0,0,2.599916056557537`,12.24370= 2660922095`,0,0,0,-2.352686800371387`,0,0,-0.7873020502993349`,0,0,0,0,0,0}= , {0,0,0,0,0,0,-1.6661117377481922`,2.857877542171938`,0,-0.557546880846985`,= 0,-4.50375057162902`,0,-2.599916056557537`,-12.243702660922095`,0,0,0,1.666= 1117377481922`,-2.857877542171938`,0,0,0,0,0,0,0,0.8654174979155087`,0,0,2.= 352686800371387`,0,0,0.557546880846985`,0,4.50375057162902`,0,0,0,-0.865417= 4979155087`,0,0,0,0,0,0.7873020502993349`,0,0,0,2.599916056557537`,12.24370= 2660922095`,0,0,0,-2.352686800371387`,0,0,-0.7873020502993349`,0,0,0,0,0,0}= , {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,= 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}};
- Follow-Ups:
- Re: Orthogonazlie with Method->"Householder"
- From: Daniel Lichtblau <danl@wolfram.com>
- Re: Orthogonazlie with Method->"Householder"