Re: 'orth' command -- question

[John B. Thoo]

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!

Thanks again.

---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>]