Re: 'orth' command -- question

[John B. Thoo]

Hello, again, James.

On Apr 16, 2012, at 11:21 PM, John B. Thoo wrote:

> Hello, James.
> 
> On Apr 16, 2012, at 11:04 PM, James Sherman Jr. wrote:
> 
>> On Tue, Apr 17, 2012 at 1:37 AM, John B. Thoo <address@hidden> wrote:
>> Hi.  I'm trying to understand the command 'orth'.
>> 
>> Example 1
>> ---------
>> octave-3.2.3:46> A = [1, 2, 2; 2, 1, 2; 2, 2, 1];
>> 
>> octave-3.2.3:47> [V, LAMBDA] = eig (A); P = orth (V)
>> P =
>> 
>>  0.62060  -0.53058   0.57735
>>  0.14920   0.80275   0.57735
>> -0.76980  -0.27217   0.57735
>> 
>> octave-3.2.3:48> P'*A*P
>> ans =
>> 
>> -1.0000e+00   2.7756e-17  -9.4369e-16
>>  1.1102e-16  -1.0000e+00   8.6042e-16
>> -8.8818e-16   7.7716e-16   5.0000e+00
>> 
>> 
>> So, it appears that 'orth' provides an orthonormal basis of eigenvectors of  
>> A.
>> 
>> Example 2
>> ---------
>> octave-3.2.3:66> A = [4, 1, 0; 1, 4, 1; 0, 1, 4];
>> octave-3.2.3:67> [V, LAMBDA] = eig (A); P = orth (V)
>> P =
>> 
>> -0.023793   0.865699   0.500000
>> -0.588348   0.392232  -0.707107
>> -0.808257  -0.310998   0.500000
>> 
>> octave-3.2.3:68> P'*A*P
>> ans =
>> 
>>  4.9791e+00  -6.5271e-01   2.2204e-16
>> -6.5271e-01   4.4351e+00   9.9920e-16
>>  4.4409e-16   6.1062e-16   2.5858e+00
>> 
>> Now it appears that 'orth' does _not_ provide an orthonormal basis of 
>> eigenvectors of  A.
>> 
>> Why does 'orth' appear to behave differently in the two examples?
>> 
>> Thanks.
>> 
>> ---John.
>> 
>> -----------------------------------------------------------------------
>> "Ten thousand difficulties do not make one doubt....  A man may be annoyed 
>> that he cannot work out a mathematical problem ... without doubting that it 
>> admits an answer."
>> 
>> ---John Henry Newman [_Apologia_, p. 239 in Project Gutenberg's
>>  <http://www.gutenberg.org/ebooks/22088>]
>> 
>> 
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>> 
>> Hi John,
>> 
>> I'm not quite sure what you mean by an "orthonormal basis of eigenvectors of 
>> A".  Specifically in your second example, the matrix V is already 
>> orthonormal (V'*V = eye(3) and its columns are composed of eigenvectors of 
>> A), so I'm not sure what your intention is in calling orth is for.  Also, 
>> since orth just looks for an orthonormal basis for the column space of V, 
>> and since the eigenvectors found have distinct eigenvalues, they span all of 
>> R^3, thus any orthonormal basis for R^3 will suffice.  So, I'd say that it 
>> was just luck involved that the first case worked as you had expected and 
>> the other case didn't, and it just depends on the particular numerical 
>> algorithm involved.
> 
> Thanks for your reply.
> 
> My question was really whether one could use 'orth' "blindly," that is, 
> without having to check first if 'eig' already returned an orthonormal set of 
> eigenvectors.  If I understand correctly, one cannot avoid having to think. 
> :-)  Also, now I see that 'orth' returns an orthonormal basis that has the 
> same span only, but does not intentionally respect eigenspaces. (Right?)
> 
> The reason for seeking an orthonormal set of eigenvectors of  A is to 
> orthogonally diagonalize  A.  I'm teaching an introductory linear algebra 
> course and, for the first time, I'm assigning some problems that require the 
> use of numerical computation software.  (I figured I needed to creep into the 
> 21st century finally.)  The textbook's exercises refer to the "Dark Side," 
> but I've asked my students to use Octave.  My problem is that I'm still very 
> much a novice at using Octave, and am trying to stay 1/2 step ahead of my 
> students. :-O  So far it's been great fun!

Forgetting about eigenvectors, if given an orthogonal matrix  A,  apparently 
'orth (A)' does not necessarily return  A.  Why?  Thanks.

---John.

-----------------------------------------------------------------------
"Ten thousand difficulties do not make one doubt....  A man may be annoyed that 
he cannot work out a mathematical problem ... without doubting that it admits 
an answer."

---John Henry Newman [_Apologia_, p. 239 in Project Gutenberg's 
   <http://www.gutenberg.org/ebooks/22088>]