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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}};


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