"1^(x=defective square_matrix)" does not evaluate to I(dim(x))

[Rolf Fabian]

Dear Ben

Thanks for checking the particular defective example using matlab.

We learn that obviously matlab checks the condition number of
the eigensystem (eigenvectors as columns of v) prior to 
similarity (back-)transformation and displays a warning it
v is close to singular (which means that x is defective).

But I can't see why the magnitude of RCOND = 2.220446e-17
reported by matlab differs from
rcond = 1/cond(v) =  4.3554e-017 concluded from Octaves
result of  "cond(v) = 2.2960e+016". Does rcond in Octave and
MatLab have different meanings ? 
What does ML mean by 'badly scaled?

Thanks again

Rolf Fabian



Ben Abbott wrote:
> 
> On Monday, January 14, 2008, at 11:39AM, "Rolf Fabian"
> <Rolf.Fabian at gmx.de> wrote:
>>
>>It can be shown, that
>>1^(x = square matrix of size n) should always evaluate to
>>eye(n), no matter of the contents of exponent matrix x
>>as long as no NaN elements occur in x.
>>
>>octave-3.0.0.exe:> n=1; 1^(x = randn(n)+rand(n)*i)   # OK
>>ans =  1
>>octave-3.0.0.exe:> n=2; 1^(x = randn(n)+rand(n)*i)   # OK
>>ans =
>>   1.00000 - 0.00000i   0.00000 + 0.00000i
>>   0.00000 - 0.00000i   1.00000 + 0.00000i
>>
>>octave-3.0.0.exe:> n=3; 1^(x = randn(n)+rand(n)*i)   # OK
>>ans =
>>   1.00000 + 0.00000i   0.00000 + 0.00000i   0.00000 + 0.00000i
>>   0.00000 + 0.00000i   1.00000 + 0.00000i   0.00000 + 0.00000i
>>   0.00000 + 0.00000i   0.00000 + 0.00000i   1.00000 + 0.00000i
>>
>>octave-3.0.0.exe:> x=[0 1 0 0; 1 0 2 3; 0 0 0 1; 0 0 1 0]
>>x =
>>   0   1   0   0
>>   1   0   2   3
>>   0   0   0   1
>>   0   0   1   0
>>
>>==============================
>>octave-3.0.0.exe:> 1^x                              # FAIL
>>ans =
>>   1.00000   0.00000   1.37500   1.62500
>>   0.00000   1.00000   2.00000   1.00000
>>   0.00000   0.00000   1.00000   0.00000
>>   0.00000   0.00000   0.00000   1.00000
>>==============================
>>
>>This result differs significantly from expected result,
>>namely 'eye(4)'.
>>Hence the evaluation by Octave is definitely incorrect.
>>BTW What does MatLab report here ?
> 
> Matlab 2007b on my Mac PPC gives,
> 
>>> x=[0 1 0 0; 1 0 2 3; 0 0 0 1; 0 0 1 0];
>>> 1^x
> Warning: Matrix is close to singular or badly scaled.
>          Results may be inaccurate. RCOND = 2.220446e-17.
> 
> ans =
> 
>    1.000000000000000                   0   0.366116523516816  
> 0.366116523516816
>                    0   1.000000000000000                   0                  
> 0
>    0.000000000000000  -0.000000000000000   1.000000000000000 
> -0.000000000000000
>                    0                   0                   0  
> 1.000000000000000
> 
>>> 
> 
> Ben
>>
>>What is special with the above counter example x ?
>>At a first glance nothing, except that it has only
>>zero entries on the main diagonal and thus it is
>>traceless. It's determinant is det(x)=1, thus it
>>is regular (non-singular).
>>
>>But x is severly defective which can be checked
>>by
>>octave-3.0.0.exe:> [v,d] = eig(x)
>>v =
>>   0.70711  -0.70711  -0.70711   0.70711
>>   0.70711   0.70711  -0.70711  -0.70711
>>   0.00000  -0.00000   0.00000  -0.00000
>>   0.00000  -0.00000   0.00000   0.00000
>>
>>d =
>>   1   0   0   0
>>   0  -1   0   0
>>   0   0   1   0
>>   0   0   0  -1
>>
>>octave-3.0.0.exe:> cond(v)
>>ans = 2.2960e+016
>>octave-3.0.0.exe:> rank(v)
>>ans =  2
>>
>>There is a rank deficiency of 2 for the
>>matrix of eigenvectors v . 
>>Thus v is not invertible and the
>>obviously underlying similarity
>>transformation fails.
>>
>>=========================
>>Proposed receipe of a fix,
>> c : real or complex scalar :
>> c ^ (x = any square matrix)
>> = log ( c ) .* eye ( dimension (x) )
>>no matter about particular
>>content of matrix x ( as long
>>as input does not contain 'NaN',
>>but  -/+ Inf elements of x 
>>should be allowed / accepted.
>>=========================
>>
>>Rolf Fabian
>>
>>< r dot fabian at jacobs-university dot de>
>>
>>
>>-----
>>Rolf Fabian
>><r dot fabian at jacobs-university dot de>
>>
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>>
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-----
Rolf Fabian
<r dot fabian at jacobs-university dot de>

-- 
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